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Dissipative Physical Systems

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Dissipative Systems Analysis and Control

Part of the book series: Communications and Control Engineering ((CCE))

Abstract

In this chapter, we shall present a class of dissipative systems which correspond to models of physical systems, and hence embed in their structure the conservation of energy (first principle of thermodynamics) and the interaction with their environment through pairs of conjugated variables with respect to the power. First, we shall recall three different definitions of systems obtained by energy-based modeling: controlled Lagrangian, input–output Hamiltonian systems, and port-controlled Hamiltonian systems. We shall illustrate and compare these definitions on some simple examples.

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Change history

  • 21 May 2022

    The original version of the book was inadvertently published with incorrect text and equations. The incorrect text and equations were corrected. The erratum chapters and the book have been updated with the changes.

Notes

  1. 1.

    Carathéodory’s, Utkin’s, Aizerman–Pyatniskii’s, and Filippov’s approaches are equivalent under some conditions [46,47,48], but yield different solutions in general.

  2. 2.

    That is, the constraints are linearly independent everywhere.

  3. 3.

    Recall that \(\partial \phi (\cdot )\) is a set-valued maximum monotone operator, see Corollary 3.121.

  4. 4.

    Unbounded, because normal cones are not bounded, and unilateral, because normal cones to sets embed unilaterality.

  5. 5.

    Strictly speaking, this is true only if the function has no accumulation of discontinuities on the right, which is the case for the velocity in mechanical systems with impacts and complementarity constraints. In other words, the motion cannot “emerge” from a “reversed” accumulation of impacts, under mild assumptions on the data.

  6. 6.

    That is, all phenomena involving an infinity of events in a finite time interval, and which occur in various types of nonsmooth dynamical systems like Filippov’s differential inclusions, etc.

  7. 7.

    In Control or Robotics studies, it may be sufficient to assume that the velocity is of special bounded variation, i.e., the measure \(d\mu _{na}\) is zero. However, this measure does not hamper stability analysis as we shall see in Sect. 7.2.4, though in all rigor one cannot dispense with its presence.

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Authors and Affiliations

  1. INRIA Grenoble Rhône-Alpes, Université Grenoble Alpes, Grenoble, France

    Bernard Brogliato

  2. Centre de Recherche de Royalieu Heuristique et Diagnostic des Systèmes, Université de Technologie de Compiègne UMR-CNRS 6599, Compiègne, France

    Rogelio Lozano

  3. LAGEP, Université Lyon 1, Villeurbanne, France

    Bernhard Maschke

  4. Department of Production and Quality Engineering, Norwegian University of Science and Technology, Trondheim, Norway

    Olav Egeland

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  1. Bernard Brogliato
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  2. Rogelio Lozano
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  3. Bernhard Maschke
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  4. Olav Egeland
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Correspondence to Bernard Brogliato .

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Brogliato, B., Lozano, R., Maschke, B., Egeland, O. (2020). Dissipative Physical Systems. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-19420-8_6

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