A bilevel programming approach to the convergence analysis of control-lyapunov functions

Wentao Tang; Prodromos Daoutidis

doi:10.1109/TAC.2019.2892386

A bilevel programming approach to the convergence analysis of control-lyapunov functions

Wentao Tang, Prodromos Daoutidis

Chemical Engineering and Materials Science

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

This paper deals with the estimation of convergence rate and domain of attraction of control-Lyapunov functions in Lyapunov-based control. This pair of estimation problems has been considered only for input-affine systems with constraints on the input norm. In this paper, we propose a novel optimization framework to address the estimation of convergence rate and domain of attraction. Specifically, we formulate the estimation problems as min-max bilevel programs for the decay rate of the Lyapunov function, where the inner problem can be resolved using Karush- Kuhn-Tucker optimality conditions, and the resulting single-level programs can be transformed into and solved as mixed-integer nonlinear programs. The proposed approach is applicable to systems with input-nonaffinity or more general forms of input constraints under an input-convexity assumption.

Original language	English (US)
Article number	8610152
Pages (from-to)	4174-4179
Number of pages	6
Journal	IEEE Transactions on Automatic Control
Volume	64
Issue number	10
DOIs	https://doi.org/10.1109/TAC.2019.2892386
State	Published - Oct 2019

Bibliographical note

Funding Information:
Manuscript received April 3, 2018; revised August 20, 2018 and December 19, 2018; accepted December 30, 2018. Date of publication January 11, 2019; date of current version September 25, 2019. This work was supported by the National Science Foundation (NSF-CBET). Recommended by Associate Editor F. Mazenc. (Corresponding author: Prodromos Daoutidis.) The authors are with the Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:,tangx647@umn.edu; daout001@umn.edu).

Publisher Copyright:
© 2019 IEEE.

Keywords

Bilevel programming
Control-Lyapunov function
Domain of attraction
Lyapunov stability

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title = "A bilevel programming approach to the convergence analysis of control-lyapunov functions",

abstract = "This paper deals with the estimation of convergence rate and domain of attraction of control-Lyapunov functions in Lyapunov-based control. This pair of estimation problems has been considered only for input-affine systems with constraints on the input norm. In this paper, we propose a novel optimization framework to address the estimation of convergence rate and domain of attraction. Specifically, we formulate the estimation problems as min-max bilevel programs for the decay rate of the Lyapunov function, where the inner problem can be resolved using Karush- Kuhn-Tucker optimality conditions, and the resulting single-level programs can be transformed into and solved as mixed-integer nonlinear programs. The proposed approach is applicable to systems with input-nonaffinity or more general forms of input constraints under an input-convexity assumption.",

keywords = "Bilevel programming, Control-Lyapunov function, Domain of attraction, Lyapunov stability",

author = "Wentao Tang and Prodromos Daoutidis",

note = "Funding Information: Manuscript received April 3, 2018; revised August 20, 2018 and December 19, 2018; accepted December 30, 2018. Date of publication January 11, 2019; date of current version September 25, 2019. This work was supported by the National Science Foundation (NSF-CBET). Recommended by Associate Editor F. Mazenc. (Corresponding author: Prodromos Daoutidis.) The authors are with the Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:,tangx647@umn.edu; daout001@umn.edu). Publisher Copyright: {\textcopyright} 2019 IEEE.",

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N2 - This paper deals with the estimation of convergence rate and domain of attraction of control-Lyapunov functions in Lyapunov-based control. This pair of estimation problems has been considered only for input-affine systems with constraints on the input norm. In this paper, we propose a novel optimization framework to address the estimation of convergence rate and domain of attraction. Specifically, we formulate the estimation problems as min-max bilevel programs for the decay rate of the Lyapunov function, where the inner problem can be resolved using Karush- Kuhn-Tucker optimality conditions, and the resulting single-level programs can be transformed into and solved as mixed-integer nonlinear programs. The proposed approach is applicable to systems with input-nonaffinity or more general forms of input constraints under an input-convexity assumption.

AB - This paper deals with the estimation of convergence rate and domain of attraction of control-Lyapunov functions in Lyapunov-based control. This pair of estimation problems has been considered only for input-affine systems with constraints on the input norm. In this paper, we propose a novel optimization framework to address the estimation of convergence rate and domain of attraction. Specifically, we formulate the estimation problems as min-max bilevel programs for the decay rate of the Lyapunov function, where the inner problem can be resolved using Karush- Kuhn-Tucker optimality conditions, and the resulting single-level programs can be transformed into and solved as mixed-integer nonlinear programs. The proposed approach is applicable to systems with input-nonaffinity or more general forms of input constraints under an input-convexity assumption.

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KW - Control-Lyapunov function

KW - Domain of attraction

KW - Lyapunov stability

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A bilevel programming approach to the convergence analysis of control-lyapunov functions

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