Contraction analysis for a class of nondifferentiable systems with applications to stability and network synchronization

SIAM Journal on Control and Optimization

Volumn 52, Issue 5, 2014, Pages 3203-3227

  Di Bernardo, Mario ^a,b   Liuzza, Davide ^c   Russo, Giovanni ^a  

^a UNIVERSITY OF NAPLES FEDERICO II   (Italy)

^b UNIVERSITY OF BRISTOL   (United Kingdom)

^c ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

Author keywords

Contraction theory; Convergence; Networks; Piecewise smooth systems

Indexed keywords

ASYMPTOTIC STABILITY; LINEAR SYSTEMS; NETWORKS (CIRCUITS); SWITCHING SYSTEMS; SYSTEM STABILITY;

CONTRACTION THEORY; CONVERGENCE; EXISTENCE AND UNIQUENESS; GLOBAL ASYMPTOTIC STABILITY; NETWORK SYNCHRONIZATION; PIECEWISE SMOOTH DYNAMICAL SYSTEMS; PIECEWISE-SMOOTH SYSTEMS; SWITCHED LINEAR SYSTEM;

DYNAMICAL SYSTEMS;

EID: 84911091329     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/120883001     Document Type: Article

References (52)

1. A.F. ANDREEV AND Y.S. BOGDANOV, On continuous dependence of solution of the Cauchy problem on initial data, Upsekhi Mat. Nauk., 13 (1958), pp. 165-166.
2. D. ANGELI, A Lyapunov approach to incremental stability properties, IEEE Trans. Automat. Control, 47 (2002), pp. 410-421.
3. M. ARCAK, Certifying spatially uniform behavior in reaction-diffusion PDE and compartmental ODE systems, Automatica, 47 (2011), pp. 1219-1229.
4. D. BERTSEKAS AND J. TSITSIKLIS, Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, Upper Saddle River, NJ, 1989.
5. S. BOCCALETTI, V. LATORA, Y. MORENO, M. CHAVEZ, AND D.U. HWANG, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), pp. 175-308.
6. V.A. BOICHENKO, G.A. LEONOV, AND V. REITMANN, Dimension Theory for Ordinary Differential Equations, Teubner, Wiesbaden, 2005.
7. V. CARMONA, E. FREIRE, E. PONCE, AND F. TORRES, The continuous matching of two stable linear systems can be unstable, Discrete Contin. Dynam. Systems, 16 (2006), pp. 689-703.
8. J. CORTES, Discontinuous dynamical systems, IEEE Control Systems Mag., 28 (2008), pp. 36-73.
9. G. DAHLQUIST, Stability and error bounds in the numerical integration of ordinary differential equations, J. Appl. Math. Mech., 41 (1959), pp. 267-268.
10. B.P. DEMIDOVICH, Lectures on Stability Theory, Nauka, Moscow, 1967.
11. M. DI BERNARDO, C.J. BUDD, A.R. CHAMPNEYS, AND P. KOWALCZYK, Piecewise Smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008.
12. n, in the Proceedings of the IEEE Conference on Decision and Control 2014, Los Angeles, CA, 2014, to appear.
13. M. DI BERNARDO AND D. LIUZZA, A criterion for incremental stability of planar Filippov systems, in Proceedings of the European Control Conference, 2013, pp. 3706-3711.
14. K. EL RIFAI AND J.J.E. SLOTINE, Compositional contraction analysis of resetting hybrid nonlinear systems, IEEE Trans. Automat. Control, 51 (2006), pp. 1536-1541.
15. A.F. FILIPPOV, Differential Equations with Discontinuous Righthand Sides, Kluwer, Dordrecht, 1988.
16. F. FORNI AND R. SEPULCHRE, A differential Lyapunov framework for contraction analysis, IEEE Trans. Automat. Control, 59 (2014), pp. 614-628.
17. P. HARTMAN, On stability in the large for systems of ordinary differential equations, Canad. J. Math., 13 (1961), pp. 480-492.
18. J. JOUFFROY, Some ancestors of contraction analysis, in Proceedings of the International Conference on Decision and Control, IEEE, Piscataway, NJ, 2005, pp. 5450-5455.
19. D.C. LEWIS, Metric properties of differential equations, Amer. J. Math., 71 (1949), pp. 294-312.
20. D. LIBERZON, Switching in Systems and Control, Birkhäuser, Berlin, 2003.
21. H. LIN AND P.J. ANTSAKLIS, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Control, 54 (2009), pp. 308-322.
22. W. LOHMILLER AND J.J.E. SLOTINE, On contraction analysis for non-linear systems, Automatica J. IFAC, 34 (1998), pp. 683-696.
23. W. LOHMILLER AND J.J.E. SLOTINE, Nonlinear process control using contraction theory, AIChe J., 46 (2000), pp. 588-596.
24. W. LOHMILLER AND J.J.E. SLOTINE, Contraction analysis of non-linear distributed systems, Internat. J. Control, 78 (2005), pp. 678-688.
25. S.M. LOZINSKII, Error estimate for numerical integration of ordinary differential equations. I, Izv. Vyssh. Uchebn. Zaved Mat., 5 (1958), pp. 52-90.
26. A.N. MICHEL, D. LIU, AND L. HOU, Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems, Springer-Verlag, New York, 2008.
27. A. PAVLOV, N. POGROMSKY, N. VAN DE WOUW, AND H. NIJMEIER, On convergence properties of piecewise affine systems, Internat. J. Control, 80 (2007), pp. 1233-1247.
28. A. PAVLOV, A. POGROMVSKY, N. VAN DE WOUV, AND H. NIJMEIJER, Convergent dynamics, a tribute to Boris Pavlovich Demidovich, Systems Control Lett., 52 (2004), pp. 257-261.
29. A. PAVLOV AND N. VAN DE WOUW, Convergent discrete-time nonlinear systems: The case of PWA systems, in Proceedings of the American Control Conference, IEEE, Piscataway, NJ, 2008, pp. 3452-3457.
30. A. PAVLOV, N. VAN DEWOUW, AND H. NIJMEIER, Convergent piecewise affine systems: Analysis and design part I: Continuous case, in Proceedings of the Conference on Decision and Control, IEEE, Piscataway, NJ, 2005, pp. 5391-5396.
31. A. PAVLOV, N. VAN DE WOUW, AND H. NIJMEIJER, Uniform Output Regulation of Nonlinear Systems: A Convergent Dynamics Approach, Birkhaüser, Basel, 2006.
32. A. PAVLOV, A. POGROMSKY, N. VON DE WOUW, H. NIJMEIJER, AND K. ROODA, Convergent piecewise affine systems: Anaysis and design part 2: Discontinuous case, in Proceedings of the Conference on Decision and Control, IEEE, Piscataway, NJ, 2005, pp. 5391-5396.
33. Q.C. PHAM AND J.J.E. SLOTINE, Stable concurrent synchronization in dynamic system networks, Neural Netw., 20 (2007), pp. 62-77.
34. M. PORFIRI, D.G. ROBERSON, AND D.J. STILWELL, Fast switching analysis of linear switched systems using exponential splitting, SIAM J. Control Optim., 47 (2008), pp. 2582-2597.
35. G. RUSSO AND M. DI BERNARDO, How to synchronize biological clocks, J. Comput. Biol., 16 (2009), pp. 379-393.
36. G. RUSSO AND M. DI BERNARDO, Solving the rendezvous problem for multi-agent systems using contraction theory, in Proceedings of the International Conference on Decision and Control, IEEE, Piscataway, NJ, 2009, pp. 5821-5826.
37. G. RUSSO AND M. DI BERNARDO, On contraction of piecewise smooth dynamical systems, in Proceedings of the 18th IFAC World Congress, Curran, Red Hook, NY, 2011.
38. G. RUSSO, M. DI BERNARDO, AND J.J. SLOTINE, A graphical algorithm to prove contraction of nonlinear circuits and systems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 58 (2011), pp. 336-348.
39. G. RUSSO, M. DI BERNARDO, AND E.D. SONTAG, Global entrainment of transcriptional systems to periodic inputs, PLoS Comput. Biol., 6 (2010), e1000739.
40. G. RUSSO, M. DI BERNARDO, AND E.D. SONTAG, A contraction approach to the hierarchical analysis and design of networked systems, IEEE Trans. Automat. Control, 58 (2013), pp. 1328-1331.
41. G. RUSSO AND J.J.E. SLOTINE, Global convergence of quorum-sensing networks, Phys. Rev. E (3), 82 (2010), 041919.
42. G. Russo and J.J.E. Slotine, Symmetries, stability, and control in nonlinear systems and networks, Phys. Rev. E (3), 85 (2012), 059902.
43. D.J.W. SIMPSON, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific, Singapore, 2010.
44. J.W. SIMPSON-PORCO AND F. BULLO, Contraction theory on Riemannian manifolds, Systems Control Lett., 65 (2014), pp. 74-80.
45. J.J. SLOTINE AND W. LI, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ, 1990.
46. J.J.E. SLOTINE, W. WANG, AND K.E. RIFAI, Contraction analysis of synchronization of nonlinearly coupled oscillators, in International Symposium on Mathematical Theory of Networks and Systems, Katholieke Universiteit Leuven, Heverlee, Belgium, 2004.
47. E.D. SONTAG, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, Springer-Verlag, New York, 1998.
48. T. STRÖM, On logarithmic norms, SIAM J. Numer. Anal., 12 (1975), pp. 741-753.
49. R.A. VAN DEN BERG, A. POGROMSKY, G.A. LEONOV, and J. ROODA, Design of convergent switched systems, in Group Coordination and Cooperative Control, Springer-Verlag, Berlin, 2006, pp. 291-311.
50. M. VIDYASAGAR, Nonlinear Systems Analysis, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1993.
51. W. WANG AND J.J.E. SLOTINE, On partial contraction analysis for coupled nonlinear oscillators, Biol. Cybernet., 92 (2005), pp. 38-53.
52. V.A. YAKUBOVICH, Matrix inequalities method in stability theory for nonlinear control systems: I. Absolute stability of forced vibrations, Autom. Remote Control, 7 (1964), pp. 905-917.