Article

Approximate reachability computation for polynomial systems

Author:

Thao DangAuthors Info & Claims

HSCC'06: Proceedings of the 9th international conference on Hybrid Systems: computation and control

Pages 138 - 152

https://doi.org/10.1007/11730637_13

Published: 29 March 2006 Publication History

Abstract

In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial. We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2. In addition, we have successfully applied our reachability algorithm to two models of a biological system.

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Cited By

Titi JGarloff J(2017)Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomialsApplied Mathematics and Computation10.1016/j.amc.2017.07.026315:C(246-258)Online publication date: 15-Dec-2017
https://dl.acm.org/doi/10.1016/j.amc.2017.07.026
Dreossi TDang TPiazza C(2017)Reachability computation for polynomial dynamical systemsFormal Methods in System Design10.1007/s10703-016-0266-350:1(1-38)Online publication date: 1-Mar-2017
https://dl.acm.org/doi/10.1007/s10703-016-0266-3
Bak SBogomolov SHenzinger TJohnson TPrakash PAbate AFainekos G(2016)Scalable Static Hybridization Methods for Analysis of Nonlinear SystemsProceedings of the 19th International Conference on Hybrid Systems: Computation and Control10.1145/2883817.2883837(155-164)Online publication date: 11-Apr-2016
https://dl.acm.org/doi/10.1145/2883817.2883837
Show More Cited By

Index Terms

Approximate reachability computation for polynomial systems
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials
      2. Numerical differentiation
2. Theory of computation
  1. Design and analysis of algorithms
  2. Randomness, geometry and discrete structures

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Guide Proceedings

HSCC'06: Proceedings of the 9th international conference on Hybrid Systems: computation and control

March 2006

582 pages

ISBN:3540331700

Editors:
João P. Hespanha
Center for Control, Dynamical Systems and Computation, University of California, Santa Barbara, CA
,
Ashish Tiwari
Center for Control, Dynamical Systems and Computation, SRI International, 333 Ravenswood Ave, Menlo Park, CA, U.S.A

Sponsors

University of California, Santa Barbara
National Science Foundation, USA
SRI Intl: SRI International

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 29 March 2006

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Cited By

Titi JGarloff J(2017)Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomialsApplied Mathematics and Computation10.1016/j.amc.2017.07.026315:C(246-258)Online publication date: 15-Dec-2017
https://dl.acm.org/doi/10.1016/j.amc.2017.07.026
Dreossi TDang TPiazza C(2017)Reachability computation for polynomial dynamical systemsFormal Methods in System Design10.1007/s10703-016-0266-350:1(1-38)Online publication date: 1-Mar-2017
https://dl.acm.org/doi/10.1007/s10703-016-0266-3
Bak SBogomolov SHenzinger TJohnson TPrakash PAbate AFainekos G(2016)Scalable Static Hybridization Methods for Analysis of Nonlinear SystemsProceedings of the 19th International Conference on Hybrid Systems: Computation and Control10.1145/2883817.2883837(155-164)Online publication date: 11-Apr-2016
https://dl.acm.org/doi/10.1145/2883817.2883837
Di Cairano SHeemels MLazar MBemporad ABelta CIvančić F(2013)Hybrid control lyapunov functions for the stabilization of hybridsystemsProceedings of the 16th international conference on Hybrid systems: computation and control10.1145/2461328.2461341(73-82)Online publication date: 8-Apr-2013
https://dl.acm.org/doi/10.1145/2461328.2461341
Sassi MTestylier RDang TGirard A(2012)Reachability analysis of polynomial systems using linear programming relaxationsProceedings of the 10th international conference on Automated Technology for Verification and Analysis10.1007/978-3-642-33386-6_12(137-151)Online publication date: 3-Oct-2012
https://dl.acm.org/doi/10.1007/978-3-642-33386-6_12
Dang TTestylier RCaccamo MFrazzoli EGrosu R(2011)Hybridization domain construction using curvature estimationProceedings of the 14th international conference on Hybrid systems: computation and control10.1145/1967701.1967721(123-132)Online publication date: 12-Apr-2011
https://dl.acm.org/doi/10.1145/1967701.1967721
Dang TMaler OTestylier RJohansson KYi W(2010)Accurate hybridization of nonlinear systemsProceedings of the 13th ACM international conference on Hybrid systems: computation and control10.1145/1755952.1755956(11-20)Online publication date: 12-Apr-2010
https://dl.acm.org/doi/10.1145/1755952.1755956
Pozo FRodellar J(2010)Robust stabilisation of polynomial systems with uncertain parametersInternational Journal of Systems Science10.1080/0020772090314142641:5(575-584)Online publication date: 1-May-2010
https://dl.acm.org/doi/10.1080/00207720903141426
Zhang MZhang SGao DWang J(2007)Analysis for dynamic of analog circuits by using HSPNProceedings of the 11th Conference on Proceedings of the 11th WSEAS International Conference on Circuits - Volume 1110.5555/1348610.1348652(207-212)Online publication date: 23-Jul-2007
https://dl.acm.org/doi/10.5555/1348610.1348652

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