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Mathematical Control Theory I pp 363–384Cite as

Trajectory-Based Theory for Hybrid Systems

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 461))

Abstract

This chapter presents a trajectory-based perspective in solving safety/ reachability analysis and synthesis problems and fault diagnosability analysis in hybrid systems. The main tool used in obtaining the results presented in this chapter is the concept of trajectory robustness, which is derived from the theory of approximate bisimulation. Trajectory robustness essentially provides a guarantee on how far the system’s state trajectories can deviate (in \(L_{\infty }\) norm) as a result of initial state variations. It further leads to the possibility of approximating the set of the system’s trajectories, which is infinite, with a finite set of trajectories. This fact, in turns, allows us to pose the above problems as finitely many finite problems that can be practically solved. In addition, these finite problems can be solved in parallel.

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Notes

  1. 1.

    For simplicity, in this chapter we do not consider nondeterminism and stochasticity in the hybrid system dynamics.

  2. 2.

    We assume that Init and Unsafe do not intersect. Otherwise, the problem is trivial.

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Acknowledgments

The author wishes to acknowledge the support from the National Science Foundation through the CAREER grant CNS-0953976 and the grant CNS-1218109 for the research leading to results presented here. The results are summarized from the author’s earlier work in collaboration with George Pappas, Antoine Girard, Georgios Fainekos, Alessandro D’Innocenzo, and graduate students Sina Afshari, Andrew Winn, and Yi Deng.

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Authors and Affiliations

  1. Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, 110 Eighth Street, NY, Troy, 12180, USA

    A. Agung Julius

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  1. A. Agung Julius
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Correspondence to A. Agung Julius .

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Editors and Affiliations

  1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands

    M. Kanat Camlibel

  2. Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA

    A. Agung Julius

  3. Department of Electrical Engineering, Indian Institute of Technology, Chennai, Tamil Nadu, India

    Ramkrishna Pasumarthy

  4. Engineering and Technology Institute Groningen, University of Groningen, Groningen, The Netherlands

    Jacquelien M.A. Scherpen

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Julius, A.A. (2015). Trajectory-Based Theory for Hybrid Systems. In: Camlibel, M., Julius, A., Pasumarthy, R., Scherpen, J. (eds) Mathematical Control Theory I. Lecture Notes in Control and Information Sciences, vol 461. Springer, Cham. https://doi.org/10.1007/978-3-319-20988-3_20

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