A Caltech Library Service

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

Gonçalves, Jorge M. and Megretski, Alexandre and Dahleh, Munther A. (2003) Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions. IEEE Transactions on Automatic Control, 48 (12). pp. 2089-2106. ISSN 0018-9286. doi:10.1109/TAC.2003.820061.

See Usage Policy.


Use this Persistent URL to link to this item:


This paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems (PLS). This methodology infers global properties of PLS solely by studying the 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. Such maps are known to be "unfriendly" maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an linear time-invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. This representation of impact maps allows the search for surface Lyapunov functions (SuLF) to be done by simply solving a semidefinite program, allowing global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS to be efficiently checked. This new analysis methodology has been applied to relay feedback, on/off and saturation systems, where it has shown to be very successful in globally analyzing a large number of examples. 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. This success in globally analyzing certain classes of PLS has shown the power of this new methodology, and suggests its potential toward the analysis of larger and more complex PLS.

Item Type:Article
Related URLs:
URLURL TypeDescription
Additional Information:© Copyright 2003 IEEE. Reprinted with permission. Manuscript received June 6, 2001; revised January 7, 2003, May 3, 2003, and June 13, 2003. [Posted online: 2004-01-08] Recommended by Associate Editor A. Bemporad.
Subject Keywords:Hybrid systems, surface Lyapunov functions (SuLF), impact and Poincaré maps, global stability
Issue or Number:12
Record Number:CaltechAUTHORS:GONieeetac03
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9773
Deposited By: Archive Administrator
Deposited On:14 Mar 2008
Last Modified:08 Nov 2021 21:02

Repository Staff Only: item control page