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Numerical computation of control Lyapunov functions in the sense of generalized gradients

URN to cite this document: urn:nbn:de:bvb:703-epub-1878-1

Title data

Baier, Robert ; Hafstein, Sigurdur Freyr:
Numerical computation of control Lyapunov functions in the sense of generalized gradients.
Department of Mathematics, University of Bayreuth
Bayreuth , 2014 . - 8 S.

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Project information

Project title:
Project's official titleProject's id
Marie-Curie Initial Training Network "Sensitivity Analysis for Deterministic Controller Design" (SADCO)264735-SADCO

Project financing: Andere
European Union "FP7-People-ITN" programme

Abstract

The existence of a control Lyapunov function with the weak infinitesimal decrease via the Dini or the proximal subdifferential and the lower Hamiltonian characterizes asymptotic controllability of nonlinear control systems and differential inclusions. We study the class of nonlinear differential inclusions with a right-hand side formed by the convex hull of active C² [$C^2$] functions which are defined on subregions of the domain. For a simplicial triangulation we parametrize a control Lyapunov function (clf) for nonlinear control systems by a continuous, piecewise affine (CPA) function via its values at the nodes and demand a suitable negative upper bound in the weak decrease condition on all vertices of all simplices. Applying estimates of the proximal subdifferential via active gradients we can set up a mixed integer linear problem (MILP) with inequalities at the nodes of the triangulation which can be solved to obtain a CPA function. The computed function is a clf for the nonlinear control system. We compare this novel approach with the one applied to compute Lyapunov functions for strongly asymptotically stable differential inclusions and give a first numerical example.

Further data

Item Type: Preprint, postprint
Additional notes (visible to public): Contents:
I. Preliminaries,
II. Control Lyapunov Functions,
III. Approach with Mixed Integer Programming,
IV. Numerical Example,
V. Conclusions.

© 2014 IEEE. Reuse of this content is subject to the IEEE Copyright.
This content will be published in: Proceedings on the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014), July 7–11, 2014, University of Groningen, Groningen, Netherlands, check the forthcoming abstract in IEEE Explore.
Keywords: control Lyapunov functions; asymptotic controllability; nonlinear control systems; continuous, piecewise affine functions; mixed integer linear programming
Subject classification: Mathematics Subject Classification Code: 93D30 (93B05 90C11)
DDC Subjects: 500 Science > 510 Mathematics
Institutions of the University: Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Profile Fields
Profile Fields > Advanced Fields
Profile Fields > Advanced Fields > Nonlinear Dynamics
Language: English
Originates at UBT: Yes
URN: urn:nbn:de:bvb:703-epub-1878-1
Date Deposited: 04 Mar 2015 11:01
Last Modified: 28 Mar 2019 11:01
URI: https://epub.uni-bayreuth.de/id/eprint/1878

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