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Dissipative 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 will further study the concept of dissipative systems which is a very useful tool in the analysis and synthesis of control laws, for linear and nonlinear dynamical systems. One of the key properties of a dissipative dynamical system is that the total energy stored in the system decreases with time. Dissipativeness can be considered as an extension of PR systems to the nonlinear case. Some relationships between positive real and passive systems have been established in Chap. 2. There exist several important subclasses of dissipative nonlinear systems with slightly different properties which are important in the analysis. Dissipativity is useful in stabilizing mechanical systems like fully actuated robots manipulators [1], robots with flexible joints [2,3,4,5,6], underactuated robot manipulators, electric motors, robotic manipulation [7], learning control of manipulators [8, 9], fully actuated and underactuated satellites [10], combustion engines [11], power converters [12,13,14,15,16,17], neural networks [18,19,20,21], smart actuators [22], piezoelectric structures [23], haptic environments and interfaces [24,25,26,27,28,29,30,31,32,33], particulate processes [34], process and chemical systems [35,36,37,38,39], missile guidance [40], model helicopters [41], magnetically levitated shafts [42, 43], biological and physiological systems [44, 45], flat glass manufacture [46], and visual feedback control [47] (see Sect. 9.4 for more references). Some of these examples will be presented in the following chapters.

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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.

    Here measurable is to be taken in the physical sense, not in the mathematical one. In other words, we assume that the process is well-equipped with suitable sensors.

  2. 2.

    That is a differential-algebraic equation (DAE) with input and output.

  3. 3.

    For instance, passivity is introduced in [95, Eq. (2.3.1)], with \(\beta =0\), and stating explicitly that it is assumed that the network has zero initial stored energy.

  4. 4.

    Once again we see that the system has zero bias provided \(x(t_{0})=0\). But in general \(\beta (x(t_{0})) \not = 0\).

  5. 5.

    i.e., the points reachable from \(x^{\star }\) with an admissible controller.

  6. 6.

    An operator may here be much more general than a linear operator represented by a constant matrix \(A \in \mathbb {R}^{m \times n}\): \(x \mapsto Ax \in \mathbb {R}^{m}\). For instance, the Laplacian \(\varDelta =\sum _{i=1}^{n} \frac{\partial ^{2}}{\partial x_{i}^{2}}\), or the D’Alembertian \(\frac{\partial ^{2}}{\partial t^{2}}-\varDelta \) are operators.

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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 Systems. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-19420-8_4

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