Title data
Baier, Robert ; Braun, Philipp ; Grüne, Lars ; Kellett, Christopher M.:
Numerical construction of nonsmooth control Lyapunov functions.
In: Giselsson, Pontus ; Rantzer, Anders
(ed.):
LargeScale and Distributed Optimization. 
Cham
: Springer
,
2018
.  pp. 343373
.  (Lecture Notes in Mathematics
; 2227
)
ISBN 9783319974774
DOI: https://doi.org/10.1007/9783319974781_12
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Project information
Project title: 
Project's official title Project's id Activating LyapunovBased Feedback  Nonsmooth Control Lyapunov Functions G1500106 

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 twodimensional 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 twodimensional example with two inputs 12.7. Conclusions 
Keywords:  Control Lyapunov functions; Mixed integer programming; Dynamical systems 
Subject classification:  Mathematics Subject Classification Code: 93D30 (90C11 93D05 90C05) 
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 > Chair 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:  03 Sep 2020 12:21 
URI:  https://eref.unibayreuth.de/id/eprint/46318 
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Numerical construction of nonsmooth control Lyapunov functions. (deposited 21 Oct 2017 21:00)
 Numerical construction of nonsmooth control Lyapunov functions. (deposited 14 Nov 2018 07:41) [Currently Displayed]