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A modal characterization of alternating approximate bisimilarity

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Abstract

Recently, alternating transition systems are adopted to describe control systems with disturbances and their finite abstract systems. In order to capture the equivalence relation between these systems, a notion of alternating approximate bisimilarity is introduced. This paper aims to establish a modal characterization for alternating approximate bisimilarity. Based on this result, we provide a link between specifications satisfied by the samples of control systems with disturbances and their finite abstractions. Moreover, a simple example is given to illustrate the application of such link in the design of controller of control systems.

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Notes

  1. By Definition 1, it is easy to see that both \(Q_{1}\cap Q'_{1}\) and \(Q_{2}\cap Q'_{2}\) are singleton. Therefore, \((Q_{1}\cap Q'_{1})\times(Q_{2}\cap Q'_{2})\) is single.

  2. \(\sigma_{1}\sim^{\varepsilon}_{\mathit{Ag}}\sigma_{2}\) if and only if for any \(i\in\mathbb{N}\), \(\sigma_{1}[i]\sim^{\varepsilon}_{\mathit{Ag}}\sigma _{2}[i]\). Similarly, \(s_{1}\sim^{\varepsilon}_{\mathit{Ag}} s_{2}\) if and only if \(s_{1}[i]\sim^{\varepsilon}_{\mathit{Ag}} s_{2}[i]\) for any i≤max{|s 1|,|s 2|}.

  3. ⋁ i∈I φ i and ⋀ i∈I φ i can be defined as usual, where I is a finite index set.

  4. In some situation, we can just observe the sampling system rather than control system with disturbances. Then, it may be reasonable to require that the sample satisfies specifications under such controller.

  5. As usual, ||y||=max{y1,y2} for y=(y1 y2)′.

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Acknowledgements

The authors are grateful to the anonymous referees for their valuable suggestions, which have helped us to improve the presentation of the paper. In particular, the approach of linear temporal logical control is perfected with the help of the anonymous referees.

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Authors and Affiliations

  1. School of Information Science, Nanjing Audit University, Nanjing, 211815, China

    Jinjin Zhang

  2. Department of Computer Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Zhaohui Zhu

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  1. Jinjin Zhang
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  2. Zhaohui Zhu
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Correspondence to Zhaohui Zhu.

Additional information

This work received financial support of the National Natural Science of China (No. 60973045), Fok Ying-Tung Education Foundation, the NSF of Jiangsu Province (No. BK2012473), the PAPD of Jiangsu Higher Education Institutions (No. YSXKKT27), and the NSF of the Jiangsu Higher Education Institutions (No. 13KJB520012).

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Zhang, J., Zhu, Z. A modal characterization of alternating approximate bisimilarity. Form Methods Syst Des 44, 240–263 (2014). https://doi.org/10.1007/s10703-013-0201-9

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