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Networked Hybrid Dynamical Systems: Models, Specifications, and Tools

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Control Subject to Computational and Communication Constraints

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 475))

Abstract

Models, specifications, and tools for networked hybrid dynamical systems are presented. The proposed modeling framework allows the agent, the network , and the algorithms to have hybrid dynamics. Notions that properly capture key specifications for networked systems, namely, formation, synchronization, safety, and security, are provided. Tools for analysis of the closed-loop hybrid system and for the design of distributed hybrid algorithms are presented. Applications of the methods to estimation, consensus, and synchronization over complex networks are presented throughout the chapter.

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Notes

  1. 1.

    A solution to \({\mathscr {H}}^a_i\) is called maximal if it cannot be extended, i.e., it is not a (proper) truncated version of another solution. It is called complete if its domain is unbounded. A solution is Zeno if it is complete and its domain is bounded in the t direction. A solution is precompact if it is complete and bounded.

  2. 2.

    In Sect. 16.5.1, we present a model for which the continuous dynamics are active when \(v_{i,1} \geqslant 0\) due to such input being connected to a strictly decreasing timer, in which case, flows with \(v_{i,1}\) identically zero are not possible. In such a case, the set \(C_i^K\) is closed.

  3. 3.

    This attractivity notion enforces that every maximal solution to \({\mathscr {H}}\) is complete, which is a property that is not for free. Sufficient conditions guaranteeing that maximal solutions are complete are given in [23, Propositions 2.10 and 6.10]. An attractivity notion that does not require every maximal solution to be complete is given in [23, Definitions 3.6 and 7.1], which, to emphasize the potential lack of completeness, has the prefix “pre.”

  4. 4.

    The solution might be trivial though, in the sense that its domain might be just one point – otherwise, points that are neither in \(\overline{C}\) nor in D would satisfy the invariance notion vacuously.

  5. 5.

    A pair \((\widetilde{\phi },d)\) defines a solution to \(\widetilde{\mathscr {H}}\) if it satisfies its dynamics. Given a hybrid arc d, its sup norm at \((t,j)\in \mathop {\mathrm{dom}}\nolimits d\) is 

    $$ \begin{aligned} \Vert d\Vert _{(t,j)}&:= \max \left\{ ess\,sup_{(s,k)\in \mathop {\mathrm{dom}}\nolimits d\setminus \varGamma (d),s+k\leqslant t+j} | d(s,k)|, \sup _{(s,k)\in \varGamma (d),s+k\leqslant t+j} | d(s,k)| \right\} \end{aligned} $$

    where \(\varGamma (d)\) denotes the set of all \((t,j) \in \mathop {\mathrm{dom}}\nolimits d\) such that \((t,j + 1)\in \mathop {\mathrm{dom}}\nolimits d\).

  6. 6.

    Given matrices A and B, \(\mathrm{He}(A,B) = A^\top B+B^\top A\), \(A \otimes B\) defines the Kronecker product, and \(A * B\) the Khatri–Rao product. The matrix \(I_n\) is the \(n \times n\) identity matrix.

  7. 7.

    A digraph is undirected if and only if the Laplacian is symmetric. The construction of \({\widetilde{\varPsi }}\) is inspired by [35].

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Authors and Affiliations

  1. Department of Computer Engineering, University of California, Santa Cruz, CA, 95064, USA

    R. G. Sanfelice

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  1. R. G. Sanfelice
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Editors and Affiliations

  1. LAAS-CNRS, Université de Toulouse, Toulouse, France

    Sophie Tarbouriech

  2. Laboratoire des Signaux et Systèmes, CNRS, CentraleSupélec, Université Paris-Sud, Université Paris-Saclay, Gif-sur-Yvette, France

    Antoine Girard

  3. CNRS, Centrale Lille, UMR 9189—Centre de Recherche en Informatique Signal et Automatique de Lille (CRIStAL), Université Lille, Lille, France

    Laurentiu Hetel

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Sanfelice, R.G. (2018). Networked Hybrid Dynamical Systems: Models, Specifications, and Tools. In: Tarbouriech, S., Girard, A., Hetel, L. (eds) Control Subject to Computational and Communication Constraints. Lecture Notes in Control and Information Sciences, vol 475. Springer, Cham. https://doi.org/10.1007/978-3-319-78449-6_16

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  • DOI: https://doi.org/10.1007/978-3-319-78449-6_16

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