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Fast Algorithms for Computing Continuous Piecewise Affine Lyapunov Functions

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Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 873))

Abstract

Algorithms that parameterize continuous and piecewise affine Lyapunov functions for nonlinear systems, both in continuous and discrete time, have been proposed in numerous publications. These algorithms generate constraints that are linear in the values of a function at all vertices of a simplicial complex. If these constraints are fulfilled for certain values at the vertices, then they can be interpolated on the simplices to deliver a function that is a Lyapunov function for the system used for their generation. There are two different approaches to find values that fulfill the constraints. First, one can use optimization to compute appropriate values that fulfill the constraints. These algorithms were originally designed for continuous-time systems and their adaptation to discrete-time systems and control systems poses some challenges in designing and implementing efficient algorithms and data structures for simplicial complexes. Second, one can use results from converse theorems in the Lyapunov stability theory to generate good candidates for suitable values and then verify the constraints for these values. In this paper we study several efficient data structures and algorithms for these computations and discuss their implementations in C++.

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Acknowledgement

The author’s research is supported by the Icelandic Research Fund (Rannís) ‘Complete Lyapunov functions: Efficient numerical computation’ (163074-052) and ‘Lyapunov Methods and Stochastic Stability’ (152429-051), which is gratefully acknowledged.

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Authors and Affiliations

  1. University of Iceland, Dunhagi 5, 107, Reykjavik, Iceland

    Sigurdur Hafstein

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  1. Sigurdur Hafstein
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Correspondence to Sigurdur Hafstein .

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Editors and Affiliations

  1. King Abdullah II School of Information Technology, The University of Jordan, Amman, Jordan

    Mohammad S. Obaidat

  2. School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON, Canada

    Tuncer Ören

  3. Department of Informatics, Modeling, Electronics and System Engineering, University of Calabria, Rende, Italy

    Floriano De Rango

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Hafstein, S. (2019). Fast Algorithms for Computing Continuous Piecewise Affine Lyapunov Functions. In: Obaidat, M., Ören, T., Rango, F. (eds) Simulation and Modeling Methodologies, Technologies and Applications . SIMULTECH 2017. Advances in Intelligent Systems and Computing, vol 873. Springer, Cham. https://doi.org/10.1007/978-3-030-01470-4_15

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