Cited By
View all- Fribourg LGoubault EMohamed SMrozek MPutot S(2020)A topological method for finding invariant sets of continuous systemsInformation and Computation10.1016/j.ic.2020.104581(104581)Online publication date: Mar-2020
A Topological Method for Finding Invariant Sets of Switched Systems
Pages 61 - 70
Abstract
We revisit the problem of finding controlled invariants sets (viability), for a class of differential inclusions, using topological methods based on Wazewski property. In many ways, this generalizes the Viability Theorem approach, which is itself a generalization of the Lyapunov function approach for systems described by ordinary differential equations. We give a computable criterion based on SoS methods for a class of differential inclusions to have a non-empty viability kernel within some given region. We use this method to prove the existence of (controlled) invariant sets of switched systems inside a region described by a polynomial template, both with time-dependent switching and with state-based switching through a finite set of hypersurfaces. A Matlab implementation allows us to demonstrate its use.
References
[1]
W. W. Adams and P. Loustaunau. An introduction to Grobner bases. Graduate studies in mathematics. American mathematical society, 1994.
[2]
E. Asarin, O. Bournez, D. Thao, O. Maler, and A. Pnueli. Effective synthesis of switching controllers for linear systems. Proc. of the IEEE, 88(7):1011--1025, July 2000.
[3]
J.-P. Aubin and A. Cellina. Differential Inclusions, Set-Valued Maps And Viability Theory. Number 264 in Grundl. der Math. Wiss. Springer, 1984.
[4]
M. Boccadoro, Y. Wardi, M. Egerstedt, and E. Verriest. Optimal control of switching surfaces in hybrid dynamical systems. Discrete Event Dynamical Systems, 15:433--448, 2005.
[5]
P. Cardaliaguet. Conditions suffisantes de non-vacuité du noyau de viabilité. C. R. Acad. Sci. Paris Ser. I, 314:797--800, 1992.
[6]
S. Coogan and M. Arcak. Guard synthesis of hybrid systems using sum of squares programming. In Conference on Decision and Control, 2012.
[7]
A. F. Filippov. Differential equations with discontinuous righthand sides. Mathematics and its applications. Kluwer, 1988.
[8]
L. Fribourg, É. Goubault, S. Mohamed, M. Mrozek, and S. Putot. A topological method for finding invariants of continuous systems. In Workshop on Reachability Problems, volume 9328 of LNCS. Springer, 2015.
[9]
G. Gabor and M. Quincampoix. On existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite-dimensional space. Journal of Differential Equations, 185:483--512, 2002.
[10]
A. Girard and G. Pappas. Approximated bisimulation relations for constrained linear systems. Automatica, 43:1307--1317, 2007.
[11]
A. Girard, G. Pola, and P. Tabuada. Approximately bisimilar symbolic models for incrementally stable switched systems. IEEE Transactions on Automatic Control, 55:116--126, 2010.
[12]
E. Haghverdi, P. Tabuada, and G. Pappas. Bisimulation relations for dynamical, control, and hybrid systems. Theor. Comput. Sci., 342, 2005.
[13]
B. Ingalls, E. D. Sontag, and Y. Wang. An infinite-time relaxation theorem for differential inclusions. Proceedings of the AMS, 131(2), 2002.
[14]
S. Jha, S. Gulwani, S. Seshia, and A. Tiwari. Synthesizing switching logic for safety and dwell-time requirements. In ACM/IEEE Int. Conf. on Cyber-Physical Systems, 2010.
[15]
A. Julius and A. van der Schaft. The maximal controlled invariant set of switched linear systems. In Conference on Decision and Control. IEEE, 2002.
[16]
S. Lang. Algebra. Graduate Texts in Mathematics. Springer New York, 2002.
[17]
J.-B. Lasserre. Moments, positive polynomials and their applications, volume 1. World Scientific, 2009.
[18]
D. Liberzon. Switching in Systems and Control. Systems and Control: Foundations and Applications. Birkhauser, 2003.
[19]
J. Lygeros, C. Tomlin, and S. Sastry. Controllers for reachability specifications for hybrid systems. Automatica, 35:349--370, 1999.
[20]
K. Mischaikow and M. Mrozek. Conley index theory. Handbook of Dynamical Systems II, 2002.
[21]
R. Moeckel. Sturm's algorithm and isolating blocks. Journal of Symbolic Computation, 40:1242--1255, 2005.
[22]
P. Nilsson, U. Boscain, M. Sigalotti, and J. Newling. Invariant sets of defocused switched systems. In Conference of Decision and Control, 2013.
[23]
N. Ozay, J. Liu, P. Prabhakar, and R. M. Murray. Computing augmented finite transition systems to synthesize switching protocols for polynomial switched systems. In American Control Conference, ACC 2013, Washington, DC, USA, June 17--19, 2013, pages 6237--6244, 2013.
[24]
Z. Sun and S. S. Ge. Stability Theory of Switched Dynamical Systems. Springer, 2011.
[25]
C. Tomlin, J. Lygeros, and S. Sastry. A game theoretic approach to controller design for hybrid systems. In Proceedings of the IEEE, 2000.
Index Terms
- A Topological Method for Finding Invariant Sets of Switched Systems
Recommendations
Boundary-Value Problems for Systems of Hamilton--Jacobi--Bellman Inclusions with Constraints
We study in this paper boundary-value problems for systems of Hamilton--Jacobi--Bellman first-order partial differential equations and variational inequalities, the solutions of which are constrained to obey viability constraints. They are motivated by ...
A necessary condition of viability for fractional differential equations with initialization
In this paper, viability results for nonlinear fractional differential equations with the Riemann-Liouville derivative are proved. We give a necessary condition for fractional viability of a locally closed set with respect to a nonlinear function.
An algorithm to compute invariant sets for third-order switched systems
We study invariant sets for switched systems. We focus on a class of third-order switched systems. In this class of switched systems we solve the problem of finding an invariant set for a switching law. For each switched system we provide the invariant ...
Comments
Information & Contributors
Information
Published In
April 2016
324 pages
Copyright © 2016 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 the author(s) 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
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 11 April 2016
Check for updates
Author Tags
Qualifiers
- Research-article
Funding Sources
- Thales Dassault Aviation DCNS DGA Ecole polytechnique ENSTA PT Telecom PT FX FDO F ParisTech
- ANR
- Digiteo
Conference
HSCC'16
Sponsor:
HSCC'16: 19th International Conference on Hybrid Systems: Computation and Control
April 12 - 14, 2016
Vienna, Austria
Acceptance Rates
HSCC '16 Paper Acceptance Rate 28 of 65 submissions, 43%;
Overall Acceptance Rate 153 of 373 submissions, 41%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 100Total Downloads
- Downloads (Last 12 months)9
- Downloads (Last 6 weeks)1
Reflects downloads up to 02 Mar 2025
Other Metrics
Citations
Cited By
View all- Fribourg LGoubault EMohamed SMrozek MPutot S(2020)A topological method for finding invariant sets of continuous systemsInformation and Computation10.1016/j.ic.2020.104581(104581)Online publication date: Mar-2020
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in