Stabilization via symmetry switching in hybrid dynamical systems

Previous Article
A rigorous derivation of hemitropy in nonlinearly elastic rods
DCDS-B Home
This Issue
Next Article
On a class of three dimensional Navier-Stokes equations with bounded delay

July 2011, 16(1): 239-263. doi: 10.3934/dcdsb.2011.16.239

Stabilization via symmetry switching in hybrid dynamical systems

Sebastian Hage-Packhäuser ^1, and Michael Dellnitz ^1,

1.	Chair of Applied Mathematics, University of Paderborn, D-33098 Paderborn, Germany, Germany

Received June 2010 Revised February 2011 Published April 2011

Abstract
Related Papers

With a view to stabilization issues of hybrid systems exhibiting a regular structure in terms of symmetry, we introduce the concept of symmetry switching and relate symmetry-induced switching strategies to the asymptotic stability of switched linear systems. To this end, a general notion of hybrid symmetries for switched systems is established whereupon orbital switching is treated which builds on the existence of hybrid symmetries. In the main part, we formulate and prove sufficient conditions for asymptotic stability under slow symmetry switching. As an example of both theoretical and practical interest, we examine time-varying networks of dynamical systems and perform stabilization by means of orbital switching. Behind all that, this work is meant to provide the groundwork for the treatment of equivariant bifurcation phenomena of hybrid systems.

Keywords: symmetries, stability., Hybrid systems.

Mathematics Subject Classification: Primary: 34C14, 34D05, 34A30; Secondary: 34A3.

Citation: Sebastian Hage-Packhäuser, Michael Dellnitz. Stabilization via symmetry switching in hybrid dynamical systems. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 239-263. doi: 10.3934/dcdsb.2011.16.239

References:

[1]	M. Dellnitz and S. Hage-Packhäuser, A global symmetry framework for Hybrid dynamical systems,, preprint., ().
[2]	B. Dionne, M. Golubitsky and I. Stewart, Coupled cells with internal symmetry: I. Wreath products, Nonlinearity, 9 (1996), 559-574. doi: 10.1088/0951-7715/9/2/016.
[3]	B. Dionne, M. Golubitsky and I. Stewart, Coupled cells with internal symmetry: II. Direct products, Nonlinearity, 9 (1996), 575-599. doi: 10.1088/0951-7715/9/2/017.
[4]	B. Fiedler, "Global Bifurcation of Periodic Solutions with Symmetry," volume $1309$ of Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 1988.
[5]	M. Golubitsky and I. Stewart, "The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space," volume $200$ of Progress in Mathematics, Birkhäuser, Berlin, 2002.
[6]	M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory Vol. II," volume $69$ of Applied Mathematical Sciences, Springer, New York, 1988.
[7]	M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the American Mathematical Society, 43 (2006), 305-364. doi: 10.1090/S0273-0979-06-01108-6.
[8]	L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. doi: 10.1016/0024-3795(95)90006-3.
[9]	P. Holmes, J. L. Lumley and G. Berkooz, "Turbulence, Coherent Structures, Dynamical Systems and Symmetry," volume II of Applied Mathematical Sciences, Cambridge University Press, 1996.
[10]	J. S. W. Lamb, k-symmetry and return maps of spacetime symmetric flows, Nonlinearity, 11 (1998), 601-629. doi: 10.1088/0951-7715/11/3/011.
[11]	D. Liberzon, "On New Sufficient Conditions for Stability of Switched Linear Systems," Proceedings of the 2009 European Control Conference, Budapest, Hungary, Aug 2009.
[12]	D. Liberzon, "Switching in Systems and Control," Systems and Control: Foundations and Applications Birkhäuser, Boston, 2003.
[13]	J. Lygeros, K. H. Johansson, S. N. Simic, J. Zhang and S. Sastry, Dynamical properties of Hybrid automata, IEEE Transactions on Automatic Control, 48 (2003), 2-17. doi: 10.1109/TAC.2002.806650.
[14]	J. Lygeros, K. H. Johansson, S. Sastry and M. Egerstedt, On the existence of executions of Hybrid automata, Proceedings of the 38th IEEE Conference on Decision and Control 1999, 3 (1999), 2249-2254.
[15]	S. N. Simic, K. H. Johansson, J. Lygeros and S. Sastry, Towards a geometric theory of Hybrid systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 12 (2005), 649-687.
[16]	A. van der Schaft and H. Schumacher, "An Introduction to Hybrid Dynamical Systems," Number 251 in Lecture Notes in Control and Information Sciences Springer-Verlag, London, 2000.

show all references

References:

[1]	M. Dellnitz and S. Hage-Packhäuser, A global symmetry framework for Hybrid dynamical systems,, preprint., ().
[2]	B. Dionne, M. Golubitsky and I. Stewart, Coupled cells with internal symmetry: I. Wreath products, Nonlinearity, 9 (1996), 559-574. doi: 10.1088/0951-7715/9/2/016.
[3]	B. Dionne, M. Golubitsky and I. Stewart, Coupled cells with internal symmetry: II. Direct products, Nonlinearity, 9 (1996), 575-599. doi: 10.1088/0951-7715/9/2/017.
[4]	B. Fiedler, "Global Bifurcation of Periodic Solutions with Symmetry," volume $1309$ of Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 1988.
[5]	M. Golubitsky and I. Stewart, "The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space," volume $200$ of Progress in Mathematics, Birkhäuser, Berlin, 2002.
[6]	M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory Vol. II," volume $69$ of Applied Mathematical Sciences, Springer, New York, 1988.
[7]	M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the American Mathematical Society, 43 (2006), 305-364. doi: 10.1090/S0273-0979-06-01108-6.
[8]	L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. doi: 10.1016/0024-3795(95)90006-3.
[9]	P. Holmes, J. L. Lumley and G. Berkooz, "Turbulence, Coherent Structures, Dynamical Systems and Symmetry," volume II of Applied Mathematical Sciences, Cambridge University Press, 1996.
[10]	J. S. W. Lamb, k-symmetry and return maps of spacetime symmetric flows, Nonlinearity, 11 (1998), 601-629. doi: 10.1088/0951-7715/11/3/011.
[11]	D. Liberzon, "On New Sufficient Conditions for Stability of Switched Linear Systems," Proceedings of the 2009 European Control Conference, Budapest, Hungary, Aug 2009.
[12]	D. Liberzon, "Switching in Systems and Control," Systems and Control: Foundations and Applications Birkhäuser, Boston, 2003.
[13]	J. Lygeros, K. H. Johansson, S. N. Simic, J. Zhang and S. Sastry, Dynamical properties of Hybrid automata, IEEE Transactions on Automatic Control, 48 (2003), 2-17. doi: 10.1109/TAC.2002.806650.
[14]	J. Lygeros, K. H. Johansson, S. Sastry and M. Egerstedt, On the existence of executions of Hybrid automata, Proceedings of the 38th IEEE Conference on Decision and Control 1999, 3 (1999), 2249-2254.
[15]	S. N. Simic, K. H. Johansson, J. Lygeros and S. Sastry, Towards a geometric theory of Hybrid systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 12 (2005), 649-687.
[16]	A. van der Schaft and H. Schumacher, "An Introduction to Hybrid Dynamical Systems," Number 251 in Lecture Notes in Control and Information Sciences Springer-Verlag, London, 2000.

[1]	Marin Kobilarov, Jerrold E. Marsden, Gaurav S. Sukhatme. Geometric discretization of nonholonomic systems with symmetries. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 61-84. doi: 10.3934/dcdss.2010.3.61
[2]	Wei Mao, Liangjian Hu, Xuerong Mao. Razumikhin-type theorems on polynomial stability of hybrid stochastic systems with pantograph delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3217-3232. doi: 10.3934/dcdsb.2020059
[3]	Thomas I. Seidman, Olaf Klein. Periodic solutions of isotone hybrid systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 483-493. doi: 10.3934/dcdsb.2013.18.483
[4]	William Clark, Anthony Bloch, Leonardo Colombo. A Poincaré-Bendixson theorem for hybrid systems. Mathematical Control and Related Fields, 2020, 10 (1) : 27-45. doi: 10.3934/mcrf.2019028
[5]	Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149
[6]	Narciso Román-Roy. A summary on symmetries and conserved quantities of autonomous Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (3) : 541-551. doi: 10.3934/jgm.2020009
[7]	Leonardo Colombo, Fernando Jiménez, David Martín de Diego. Variational integrators for mechanical control systems with symmetries. Journal of Computational Dynamics, 2015, 2 (2) : 193-225. doi: 10.3934/jcd.2015003
[8]	Mónica Clapp, Angela Pistoia. Yamabe systems and optimal partitions on manifolds with symmetries. Electronic Research Archive, 2021, 29 (6) : 4327-4338. doi: 10.3934/era.2021088
[9]	Kathrin Flasskamp, Sebastian Hage-Packhäuser, Sina Ober-Blöbaum. Symmetry exploiting control of hybrid mechanical systems. Journal of Computational Dynamics, 2015, 2 (1) : 25-50. doi: 10.3934/jcd.2015.2.25
[10]	Andrey Tsiganov. Poisson structures for two nonholonomic systems with partially reduced symmetries. Journal of Geometric Mechanics, 2014, 6 (3) : 417-440. doi: 10.3934/jgm.2014.6.417
[11]	B. Cantó, C. Coll, A. Herrero, E. Sánchez, N. Thome. Pole-assignment of discrete time-delay systems with symmetries. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 641-649. doi: 10.3934/dcdsb.2006.6.641
[12]	Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017
[13]	Carsten Collon, Joachim Rudolph, Frank Woittennek. Invariant feedback design for control systems with lie symmetries - A kinematic car example. Conference Publications, 2011, 2011 (Special) : 312-321. doi: 10.3934/proc.2011.2011.312
[14]	Božzidar Jovanović. Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems. Journal of Geometric Mechanics, 2018, 10 (2) : 173-187. doi: 10.3934/jgm.2018006
[15]	Yanli Han, Yan Gao. Determining the viability for hybrid control systems on a region with piecewise smooth boundary. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 1-9. doi: 10.3934/naco.2015.5.1
[16]	Leonardo J. Colombo, María Emma Eyrea Irazú, Eduardo García-Toraño Andrés. A note on Hybrid Routh reduction for time-dependent Lagrangian systems. Journal of Geometric Mechanics, 2020, 12 (2) : 309-321. doi: 10.3934/jgm.2020014
[17]	Fatiha Alabau-Boussouira, Piermarco Cannarsa, Roberto Guglielmi. Indirect stabilization of weakly coupled systems with hybrid boundary conditions. Mathematical Control and Related Fields, 2011, 1 (4) : 413-436. doi: 10.3934/mcrf.2011.1.413
[18]	David L. Russell. Control via decoupling of a class of second order linear hybrid systems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1321-1334. doi: 10.3934/dcdss.2014.7.1321
[19]	Kobamelo Mashaba, Jianxing Li, Honglei Xu, Xinhua Jiang. Optimal control of hybrid manufacturing systems by log-exponential smoothing aggregation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1711-1719. doi: 10.3934/dcdss.2020100
[20]	Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633-659. doi: 10.3934/mbe.2007.4.633

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Sebastian Hage-Packhäuser Michael Dellnitz
on Google Scholar
Sebastian Hage-Packhäuser Michael Dellnitz

[Back to Top]