Titlebar

Export bibliographic data
Literature by the same author
plus on the publication server
plus at Google Scholar

 

Numerical construction of nonsmooth control Lyapunov functions

Title data

Baier, Robert ; Braun, Philipp ; Grüne, Lars ; Kellett, Christopher M.:
Numerical construction of nonsmooth control Lyapunov functions.
In: Giselsson, Pontus ; Rantzer, Anders (ed.): Large-Scale and Distributed Optimization. - Cham : Springer , 2018 . - pp. 343-373 . - (Lecture Notes in Mathematics ; 2227 )
ISBN 978-3-319-97477-4
DOI: https://doi.org/10.1007/978-3-319-97478-1_12

This is the latest version of this item.

Related URLs

Project information

Project title:
Project's official titleProject's id
Activating Lyapunov-Based Feedback - Nonsmooth Control Lyapunov FunctionsG1500106

Project financing: ARC (Australian Research Council)

Abstract in another language

Abstract Lyapunov’s second method is one of the most successful tools for analyzing stability properties of dynamical systems. If a control Lyapunov function is known, 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 Artstein’s circles and on a two-dimensional system with two inputs. The underlying optimization problems are solved in Gurobi.

Further data

Item Type: Article in a book
Refereed: Yes
Additional notes: Contents:
12.1. Introduction
12.2. Mathematical setting
12.3. Continuous piecewise affine functions
12.3.1 Discretization of the state space
12.3.2 Continuous piecewise affine functions
12.4. The decrease condition of control Lyapunov functions
12.4.1 The decrease condition for piecewise affine functions
12.4.2 Semiconcavity conditions
12.4.3 Local minimum condition
12.4.4 A finite dimensional optimization problem
12.5. Reformulation as mixed integer linear programming problem
12.5.1 Approximation of system parameters and reformulation of nonlinear constraints
12.5.2 The mixed integer linear programming formulation
12.6. Numerical examples
12.6.1 Artstein's circles
12.6.2 A two-dimensional example with two inputs
12.7. Conclusions
Keywords: control Lyapunov functions; mixed integer programming; dynamical systems
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) > Chair Mathematics V (Applied Mathematics) - Univ.-Prof. Dr. Lars Grüne)
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Professorship Applied Mathematics (Applied Mathematics)
Profile Fields > Advanced Fields > Nonlinear Dynamics
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
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 14 Nov 2018 07:41
Last Modified: 22 Nov 2018 08:38
URI: https://eref.uni-bayreuth.de/id/eprint/46318

Available Versions of this Item