Numerical construction of nonsmooth control lyapunov functions

Robert Baier, Philipp Braun, Lars Grune, Christopher Kellett

    Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

    6 Citations (Scopus)

    Abstract

    Lyapunovs second method is one of the most successful tools for analyzing stability properties of dynamical systems. If a control Lyapunov function is known, then asymptotic stabilizability of an equilibrium of the corresponding dynamical system can be concluded without the knowledge of an explicit solution of the dynamical system. Whereas necessary and sufficient conditions for the existence of nonsmooth control Lyapunov functions are known by now, constructive methods to generate control Lyapunov functions for given dynamical systems are not known up to the same extent. In this paper we build on previous work to compute (control) Lyapunov functions based on linear programming and mixed integer linear programming. In particular, we propose a mixed integer linear program based on a discretization of the state space where a continuous piecewise affine control Lyapunov function can be recovered from the solution of the optimization problem. Different to previous work, we incorporate a semiconcavity condition into the formulation of the optimization problem. Results of the proposed scheme are illustrated on the example of Artsteins circles and on a two-dimensional system with two inputs. The underlying optimization problems are solved in Gurobi (2016, http://www.gurobi.com).
    Original languageEnglish
    Title of host publicationLarge-Scale and Distributed Optimization
    EditorsPontus Giselsson, Anders Rantzer
    Place of PublicationSwitzerland
    PublisherSpringer Cham
    Pages343-373
    Volume1
    Edition1st
    ISBN (Print)978-3-319-97477-4
    DOIs
    Publication statusPublished - 2018

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