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Positive Stabilization of a Diffusion System by Nonnegative Boundary Control

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Positive Systems (POSTA 2016)

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 471))

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Abstract

This chapter deals with the issue of considering nonnegative inputs in the positive stabilization problem. It is shown in two different ways why one cannot expect to positively stabilize a positive system by use of a nonnegative input, first by a classical approach with a formal proof, then by working on an extended system for which the new input corresponds to the time derivative of the nominal one, thus circumventing the sign restriction. However, it is shown via a classical example of positive system—the pure diffusion system—that positively stabilizing a positive system with a nonnegative input is in some way possible: using a boundary control, the input sign depends on whether the boundary control appears in the boundary conditions or in the dynamics. The chapter then provides a parameterization of all positively stabilizing feedbacks for a discretized model of the pure diffusion system, some numerical simulations and a convergence discussion which allows to extend the results to the infinite-dimensional case, where the system is described again by a parabolic partial differential equation and the input acts either in the dynamics or in the boundary conditions.

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Acknowledgements

This chapter presents research results of the Belgian network DYSCO (Dynamical Systems, Control and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian state, Science Policy Office (BELSPO). The scientific responsibility rests with its authors.

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Authors and Affiliations

  1. Department of Mathematics and naXys, University of Namur, Rempart de la Vierge 8, 5000, Namur, Belgium

    Jonathan N. Dehaye & Joseph J. Winkin

Authors
  1. Jonathan N. Dehaye
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  2. Joseph J. Winkin
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Correspondence to Jonathan N. Dehaye .

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Editors and Affiliations

  1. Università Campus Biomedico , Rome, Italy

    Filippo Cacace

  2. Sapienza Università di Roma , Rome, Italy

    Lorenzo Farina

  3. Università Campus Biomedico , Rome, Italy

    Roberto Setola

  4. Università dell’Aquila , L'Aquila, Italy

    Alfredo Germani

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Dehaye, J.N., Winkin, J.J. (2017). Positive Stabilization of a Diffusion System by Nonnegative Boundary Control. In: Cacace, F., Farina, L., Setola, R., Germani, A. (eds) Positive Systems . POSTA 2016. Lecture Notes in Control and Information Sciences, vol 471. Springer, Cham. https://doi.org/10.1007/978-3-319-54211-9_14

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  • DOI: https://doi.org/10.1007/978-3-319-54211-9_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-54210-2

  • Online ISBN: 978-3-319-54211-9

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