research-article

Hybridization domain construction using curvature estimation

Authors:

Romain TestylierAuthors Info & Claims

HSCC '11: Proceedings of the 14th international conference on Hybrid systems: computation and control

Pages 123 - 132

https://doi.org/10.1145/1967701.1967721

Published: 12 April 2011 Publication History

Abstract

This paper is concerned with the reachability computation for non-linear systems using hybridization. The main idea of hybridization is to approximate a non-linear vector field by a piecewise-affine one. The piecewise-affine vector field is defined by building around the set of current states of the system a simplicial domain and using linear interpolation over its vertices. To achieve a good time-efficiency and accuracy of the reachability computation on the approximate system, it is important to find a simplicial domain which, on one hand, is as large as possible and, on the other hand, guarantees a small interpolation error. In our previous work[8], we proposed a method for constructing hybridization domains based on the curvature of the dynamics and showed how the method can be applied to quadratic systems. In this paper we pursue this work further and present two main results. First, we prove an optimality property of the domain construction method for a class of quadratic systems. Second, we propose an algorithm of curvature estimation for more general non-linear systems with non-constant Hessian matrices. This estimation can then be used to determine efficient hybridization domains. We also describe some experimental results to illustrate the main ideas of the algorithm as well as its performance.

References

[1]

M. Althoff, O. Stursberg, and M. Buss. Reachability Analysis of Nonlinear Systems with Uncertain Parameters using Conservative Linearization. CDC'08, 2008.

[2]

E. Asarin, O. Bournez, T. Dang, and O. Maler. Approximate reachability analysis of piecewise linear dynamical systems. HSCC'00, LNCS 1790, 21--31, 2000.

Digital Library

[3]

E. Asarin, T. Dang, and A. Girard. Reachability Analysis of Nonlinear Systems Using Conservative Approximation. HSCC'03, LNCS 2623, pp 20--35, Springer, 2003.

Digital Library

[4]

E. Asarin, T. Dang, and A. Girard. Hybridization Methods for the Analysis of Nonlinear Systems. Acta Informatica 43, 451--476, 2007.

Digital Library

[5]

A. Chutinan and B. H. Krogh. Computational Techniques for Hybrid System Verification. IEEE Trans. on Automatic Control 48, 64--75, 2003.

[6]

T. Dang, C. Le Guernic and O. Maler. Computing Reachable States for Nonlinear Biological Models. Computational Methods in Systems Biology CMSB'09, LNCS 5688, pp 126--141, 2009.

Digital Library

[7]

T. Dang. Approximate Reachability Computation for Polynomial Systems. HSCC'06, LNCS 3927, pp 138--152, 2006.

Digital Library

[8]

T. Dang, O. Maler, and R. Testylier. Accurate hybridization of nonlinear systems. HSCC'10, ACM, pp 11--20, 2010.

Digital Library

[9]

J. Della Dora, A. Maignan, M. Mirica-Ruse, and S. Yovine. Hybrid computation. Proceedings International Symposium on Symbolic and Algebraic Computation ISSAC, 2001.

Digital Library

[10]

A. Girard. Reachability of Uncertain Linear Systems using Zonotopes. HSCC'05, LNCS 3414, 291--305, 2005.

Digital Library

[11]

A. Girard, C. Le Guernic and O. Maler. Efficient Computation of Reachable Sets of Linear Time-invariant Systems with Inputs. HSCC'06, LNCS 3927, 257--271 2006.

Digital Library

[12]

M. R. Greenstreet and I. Mitchell, Reachability Analysis Using Polygonal Projections. HSCC'99 LNCS 1569, 103--116, 1999.

Digital Library

[13]

S.G. Johnson. The NLopt nonlinear optimization package. http://ab-initio.mit.edu/nlopt

[14]

M. Kloetzer and C. Belta, Reachability analysis of multi-affine systems. HSCC Hybrid Systems: Computation and Control, vol 3927 in LNCS, 348--362, Springer, 2006.

Digital Library

[15]

A. Kurzhanskiy and P. Varaiya, Ellipsoidal Techniques for Reachability Analysis of Discrete-time Linear Systems, IEEE Trans. Automatic Control 52, 26--38, 2007.

[16]

M. Kvasnica, P. Grieder, M. Baotic and M. Morari. Multi-Parametric Toolbox (MPT) HSCC'04, LNCS 2993, pp 448--462, Springer, 2004.

[17]

I. Mitchell and C. Tomlin. Level Set Methods for Computation in Hybrid Systems, HSCC'00, LNCS 1790, 310--323, 2000.

Digital Library

[18]

S. Sastry. Nonlinear Systems: Analysis, Stability, and Control. Springer-Verlag, 1999.

[19]

J. R. Shewchuk. What Is a Good Linear Element? Interpolation, Conditioning, and Quality Measures. Eleventh International Meshing Roundtable (Ithaca, New York), pp 115--126, Sandia National Laboratories, September 2002.

[20]

A. Tiwari and G. Khanna. Nonlinear Systems: Approximating Reach Sets. HSCC'04, LNCS 2993, pp 600--614, 2004.

[21]

P. Varaiya, Reach Set computation using Optimal Control, KIT Workshop, Verimag, Grenoble, 377--383, 1998.

Cited By

Edwards AGiacobbe MAbate A(2023)On the Trade-Off Between Efficiency and Precision of Neural AbstractionQuantitative Evaluation of Systems10.1007/978-3-031-43835-6_12(152-171)Online publication date: 15-Sep-2023
https://doi.org/10.1007/978-3-031-43835-6_12
Chen XSankaranarayanan S(2022)Reachability Analysis for Cyber-Physical Systems: Are We There Yet?NASA Formal Methods10.1007/978-3-031-06773-0_6(109-130)Online publication date: 20-May-2022
https://doi.org/10.1007/978-3-031-06773-0_6
Bartocci EDeshmukh JGigler FMateis CNickovic DQin X(2020)Mining Shape Expressions from Positive ExamplesIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems10.1109/TCAD.2020.3012240(1-1)Online publication date: 2020
https://doi.org/10.1109/TCAD.2020.3012240
Show More Cited By

Index Terms

Hybridization domain construction using curvature estimation
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations

Recommendations

Predicate abstraction for reachability analysis of hybrid systems

Embedded systems are increasingly finding their way into a growing range of physical devices. These embedded systems often consist of a collection of software threads interacting concurrently with each other and with a physical, continuous environment. ...
Verification of logic controllers for continuous plants using timed condition/event-system models

An approach to the formal verification of logic controllers for processes with switched continuous dynamics is presented. The method builds on modular, timed discrete event models of the plant and the controller. Subsystems with continuous dynamics are ...
Symbolic algorithm analysis of rectangular hybrid systems
TAMC'08: Proceedings of the 5th international conference on Theory and applications of models of computation

This paper investigates symbolic algorithm analysis of rectangular hybrid systems. To deal with the symbolic reachability problem, a restricted constraint system called hybrid zone is formalized. Hybrid zones are also applied to a symbolic model-...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

HSCC '11: Proceedings of the 14th international conference on Hybrid systems: computation and control

April 2011

330 pages

ISBN:9781450306294

DOI:10.1145/1967701

General Chair:
Marco Caccamo
University of Illinois at Urbana-Champaign, USA
,
Program Chairs:
Emilio Frazzoli
Massachusetts Institute of Technology, USA
,
Radu Grosu
State University of New York at Stony Brook, USA

Copyright © 2011 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGBED: ACM Special Interest Group on Embedded Systems

In-Cooperation

IEEE

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 12 April 2011

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

HSCC '11

Sponsor:

SIGBED

HSCC '11: Hybrid Systems: Computation and Control

April 12 - 14, 2011

IL, Chicago, USA

Acceptance Rates

Overall Acceptance Rate 153 of 373 submissions, 41%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

9
Total Citations
View Citations
100
Total Downloads

Downloads (Last 12 months)2
Downloads (Last 6 weeks)0

Reflects downloads up to 26 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Edwards AGiacobbe MAbate A(2023)On the Trade-Off Between Efficiency and Precision of Neural AbstractionQuantitative Evaluation of Systems10.1007/978-3-031-43835-6_12(152-171)Online publication date: 15-Sep-2023
https://doi.org/10.1007/978-3-031-43835-6_12
Chen XSankaranarayanan S(2022)Reachability Analysis for Cyber-Physical Systems: Are We There Yet?NASA Formal Methods10.1007/978-3-031-06773-0_6(109-130)Online publication date: 20-May-2022
https://doi.org/10.1007/978-3-031-06773-0_6
Bartocci EDeshmukh JGigler FMateis CNickovic DQin X(2020)Mining Shape Expressions from Positive ExamplesIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems10.1109/TCAD.2020.3012240(1-1)Online publication date: 2020
https://doi.org/10.1109/TCAD.2020.3012240
Bartocci ELió P(2016)Computational Modeling, Formal Analysis, and Tools for Systems BiologyPLOS Computational Biology10.1371/journal.pcbi.100459112:1(e1004591)Online publication date: 21-Jan-2016
https://doi.org/10.1371/journal.pcbi.1004591
Prabhakar PSoto MLal R(2016)Verification Techniques for Hybrid SystemsLeveraging Applications of Formal Methods, Verification and Validation: Discussion, Dissemination, Applications10.1007/978-3-319-47169-3_61(833-842)Online publication date: 5-Oct-2016
https://doi.org/10.1007/978-3-319-47169-3_61
Maler O(2014)Algorithmic Verification of Continuous and Hybrid SystemsElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.140.4140(48-69)Online publication date: 23-Feb-2014
https://doi.org/10.4204/EPTCS.140.4
Goubault EMullier OPutot SKieffer MFráänzle MLygeros J(2014)Inner approximated reachability analysisProceedings of the 17th international conference on Hybrid systems: computation and control10.1145/2562059.2562113(163-172)Online publication date: 15-Apr-2014
https://dl.acm.org/doi/10.1145/2562059.2562113
Testylier RDang T(2013)NLTOOLBOX: A Library for Reachability Computation of Nonlinear Dynamical SystemsAutomated Technology for Verification and Analysis10.1007/978-3-319-02444-8_37(469-473)Online publication date: 2013
https://doi.org/10.1007/978-3-319-02444-8_37
Testylier RDang T(2012)Analysis of parametric biological models with non-linear dynamicsElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.92.292(16-29)Online publication date: 15-Aug-2012
https://doi.org/10.4204/EPTCS.92.2

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten