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Container of (min, +)-linear systems

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Abstract

Based on the (min,+)-linear system theory, the work developed here takes the set membership approach as a starting point in order to obtain a container for ultimately pseudo-periodic functions representative of Discrete Event Dynamic Systems. Such a container, by approximating the exact system, ensures to entirely include it in a guaranteed way. To reach that point, the container introduced in this paper is given as an interval, the bounds of which are a convex function for the upper approximation and a concave function for the lower approximation. Thanks to the characteristics of the bounds, the aim is both to reduce data storage (that can be very high when exact functions are handled) and to reduce the algorithm complexity of the operations of sum, inf-convolution and subadditive closure. These operations are integrated into inclusion functions, the algorithms of which are of linear or quasi-linear complexity.

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Notes

  1. Name of a research project dealing with COmputational Issues in Network Calculus.

  2. Operation used for the concatenation of systems.

  3. Also called Kleene star operation and used for systems with closed-loop architecture.

  4. Also called the convex conjugate function.

  5. Created with the container and the algorithms described in this paper.

  6. Which can be seen as “signals” for DEDS.

  7. Subclass of Timed Petri Nets in which each place has exactly one upstream and one downstream transition.

  8. As in the usual algebra, operator ⊗ can be omitted: ab = a ⊗ b.

  9. This designation is different according to the context of use: Network Calculus or TEG.

  10. Which is made in the MinMaxGD toolbox.

  11. The affine parts are linked by non-differentiable points.

  12. The epigraph of a convex function is a convex set.

  13. The hypograph of a concave function is a convex set.

  14. \(\mathcal{D}_{convex}\) is the set of convex functions endowed with the pointwise maximum as sum and the pointwise addition as product.

  15. On ]τ f , + ∞ [.

  16. These designations come from the Network Calculus.

  17. The sorting of the function slopes is assumed to be made by the data structure used for their representation.

  18. The construction of a container f from one elementary function \(\Delta_{T}^{K}\) provides two identical bounds \(\underline{f} = \overline{f}\) with only one extremal point of (T,K) coordinates, and an infinite asymptotic slope \(\sigma(\underline{f}) = \sigma(\overline{f}) = +\infty\).

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  1. Laboratoire d’Ingénierie des Systèmes Automatisés, Université d’Angers, 62, Avenue Notre Dame du Lac, 49000, Angers, France

    Euriell Le Corronc, Bertrand Cottenceau & Laurent Hardouin

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  1. Euriell Le Corronc
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  2. Bertrand Cottenceau
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  3. Laurent Hardouin
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Correspondence to Euriell Le Corronc.

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Le Corronc, E., Cottenceau, B. & Hardouin, L. Container of (min, +)-linear systems. Discrete Event Dyn Syst 24, 15–52 (2014). https://doi.org/10.1007/s10626-012-0148-9

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