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Global analysis of piecewise linear systems using impact maps and quadratic surface Lyapunov functions

Gonçalves, Jorge M. and Megretski, Alexandre and Dahleh, Munther A. (2001) Global analysis of piecewise linear systems using impact maps and quadratic surface Lyapunov functions. In: European Control Conference (ECC), 2001. Faculdade de Engenharia da Universidade do Porto , Porto, Portugal, pp. 1357-1362. ISBN 978-3-9524173-6-2. https://resolver.caltech.edu/CaltechAUTHORS:20150513-065209753

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Abstract

In this paper we develop an entirely new constructive global analysis methodology for a class of hybrid systems known as Piecewise Linear Systems (PLS). This methodology consists in inferring global properties of PLS solely by studying their behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface, by constructing quadratic Lyapunov functions on switching surfaces. We found that an impact map induced by an LTI flow between two switching surfaces can be represented as a linear transformation analytically parameterized by a scalar function of the state. This representation of impact maps allows the search for quadratic surface Lyapunov functions to be done by simply solving a set of LMIs. Global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS can this way be efficiently checked. These new results were successfully applied to certain classes of PLS. Although this analysis methodology yields only sufficient criteria of stability, it has shown to be very successful in globally analyzing a large number of examples with a locally stable limit cycle or equilibrium point. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7076105PublisherArticle
Additional Information:© 2001 This research was supported in part by the Portuguese Science and Technology Foundation under the program “PRAXIS XXI”, by the NSF under grants ECS-9410531, ECS-9796099, ECS-9796033, and ECS-9612558, and by the AFOSR under grants F49620-96-1-0123, F49620-00-1-0096, F49620-95-0219, and F49620-99-1-0320.
Funders:
Funding AgencyGrant Number
Portuguese Science and Technology FoundationUNSPECIFIED
NSFECS-9410531
NSFECS-9796099
NSFECS-9796033
NSFECS-9612558
Air Force Office of Scientific Research (AFOSR)F49620-96-1-0123
Air Force Office of Scientific Research (AFOSR)F49620-00-1-0096
Air Force Office of Scientific Research (AFOSR)F49620-95- 0219
Air Force Office of Scientific Research (AFOSR)F49620-99-1-0320
Subject Keywords:hybrid systems, stability analysis, robustness analysis, performance analysis, switching surfaces
Record Number:CaltechAUTHORS:20150513-065209753
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20150513-065209753
Official Citation:Gonccalves, Jorge M.; Megretski, Alexandre; Dahleh, Munther A., "Global analysis of piecewise linear systems using impact maps and quadratic surface Lyapunov functions," Control Conference (ECC), 2001 European , vol., no., pp.1357,1362, 4-7 Sept. 2001
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:57471
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:13 May 2015 17:43
Last Modified:03 Oct 2019 08:25

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