On convergence properties of piecewise affine systems

International Journal of Control

Volumn 80, Issue 8, 2007, Pages 1233-1247

  Pavlov, A ^a   Pogromsky, A ^b   Van De Wouw, N ^b   Nijmeijer, H ^b  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; BIFURCATION (MATHEMATICS); CAPACITORS; CONVERGENCE OF NUMERICAL METHODS; LYAPUNOV FUNCTIONS; NONLINEAR SYSTEMS; NUMERICAL METHODS; SWITCHING;

CONTROLLER DESIGNS; PIECEWISE AFFINE SYSTEMS; QUADRATIC CONVERGENCE; SWITCHING CAPACITORS;

PIECEWISE LINEAR TECHNIQUES;

EID: 34447136184     PISSN: 00207179     EISSN: 13665820     Source Type: Journal    
DOI: 10.1080/00207170701261978     Document Type: Article

References (44)

1. D. Angeli, "A Lyapunov approach to incremental stability properties", IEEE Trans. Autom. Contr., 47, pp. 410-421, 2002.
2. J.T.J. Aprille and T. Trick, "Steady-state analysis of non-linear circuits with periodic inputs", Proc. IEEE, 60, pp. 108-114, 1972.
3. A. Bemporad, G. Ferrari-Trecate and M. Morari, "Observability and controllability of piecewise affine and hybrid systems", IEEE Trans. Autom. Contr., 45, pp. 1864-1876, 2000.
4. S. Boyd, A. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Control Theory, 15 of Studies in Applied Mathematics, SIAM, 1994.
5. 2)ẏ + y = bλk, cos(λt + a), k large", J. London Math. Soc., 20, pp. 180-189, 1945.
6. L. Chua and D. Green, "A qualitative analysis of the behavior of dynamic nonlinear networks: steady-state solutions of nonautonomous networks", IEEE Trans. Circ. Syst., 23, pp. 530-550, 1976.
7. B.P. Demidovich, "Dissipativity of a nonlinear system of differential equations, part I", Vestnik Moscow State Univiersity, Ser. Matem. Mekh., 6, pp. 19-27, 1961. (in Russian)
8. B.P. Demidovich, Lectures on Stability Theory, Moscow: Nauka, 1967, (in Russian).
9. R. Fey, "Steady-state behaviour of reduced dynamic systems with local nonlinearities". PhD thesis, Eindhoven University of Technology, The Netherlands (1992).
10. R. Fey, D.H. van Campen and A. de Kraker, "Long term structural dynamics of mechanical systems with local nonlinearities", Trans. ASME, J. Vibration and Acoustics, 118, pp. 147-153,
11. also publ. in Proc. Winter Annual Meeting ASME, Anaheim, Calif, USA, 8-13 Nov., DE-Vol. 50, AMD-Vol. 44, R.A. Ibrahim and N.S. Namachchivaya, Eds, 1992, pp. 159-167, 1996.
12. A. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and Its Applications, Dordrecht: Kluwer Academic, 1988.
13. V. Fromion, S. Monaco and D. Normand-Cyrot, "Asymptotic properties of incrementally stable systems", IEEE Trans. Autom. Contr., 41, pp. 721-723, 1996.
14. V. Fromion, G. Scorletti and G. Ferreres, "Nonlinear performance of a PI controlled missile: an explanation", Int. J. Robust Nonlinear Contr., 9, pp. 485-518, 1999.
15. I. Hajj and S. Skelboe, "Steady-state analysis of piecewise-linear dynamic systems", IEEE Trans. Circ. Syst., CAT-28, pp. 234-242, 1981.
16. W. Heemels, M. Camlibel and J. Schumacher, "On the dynamic analysis of piecewise-linear networks", IEEE Trans. Circ Syst. I: Fund. Theory Appl., 49, pp. 315-327, 2002.
17. M. Heertjes, "Controlled stabilization of long-term solutions in a piece-wise linear beam system". PhD thesis, Eindhoven University of Technology (1999).
18. M. Heertjes, M. van de Molengraft and H. Nijmeijer, "Partial feedback linearization on a harmonically excited beam with one-sided spring", J. Sound Vibr., 228, pp. 939-959, 1999.
19. M. Heertjes, N. van de Wouw, E. Pastink, A. Pavlov, H. Nijmeijer and M. Steinbuch, "Performance of variable-gain optical storage drives", in Proc. of American Control Conference, Minneapolis, USA, 2006, pp. 1976-1981.
20. M. Johansson, Piecewise Linear Control Systems - A Computational Approach, Lecture Notes in Control and Information Sciences No. 284, Heidelberg: Springer-Verlag, 2002.
21. M. Johansson and A. Rantzer, "Computation of piecewise quadratic Lyapunov functions for hybrid systems", IEEE Trans. Autom. Contr., 43, pp. 555-559, 1998.
22. A. Juloski, "Observer design and identification methods for hybrid systems: theory and experiments". PhD thesis, Eindhoven University of Technology (2004).
23. A. Juloski, M. Heemels and S. Weiland, "Observer design for a class of piecewise affine systems", in Proc. of IEEE Conf. Decision and Control, Las Vegas, USA, 2002, pp. 2606-2611.
24. J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method with Applications, New York: Academic press, 1961.
25. W. Lohmiller and J.J. Slotine, "On contraction analysis for nonlinear systems", Automatica, 34, pp. 683-696, 1998.
26. T. Parker and L. Chua, Practical Numerical Algorithms for Chaotic Systems, New York: Springer-Verlag, 1989.
27. A. Pavlov, A. Pogromsky, N. van de Wouw and H. Nijmeijer, "Convergent dynamics, a tribute to Boris Pavlovich Demidovich", Syst. Contr. Lett., 52, pp. 257-261, 2004.
28. A. Pavlov, N. van de Wouw and H. Nijmeijer, "Convergent systems: analysis and design", in Lecture Notes in Control and Information Sciences, T. Meurer, K. Graichen and D. Gilles, Eds, Vol. 322, Springer, 2005a, pp. 131-146.
29. A. Pavlov, N. van de Wouw and H. Nijmeijer, Uniform Output Regulation of Nonlinear Systems: A Convergent Dynamics Approach, Boston: Birkhauser, 2005b.
30. A. Pavlov, N. van de Wouw and H. Nijmeijer, "Frequency response functions and Bode plots for nonlinear convergent systems", in Proc of IEEE Conf. Decision Control, San Diego, USA, 2006, pp. 3765-3770.
31. V. A. Pliss, Nonlocal Problems of the Theory of Oscillations, London: Academic Press, 1966.
32. E. Sontag, "On the input-to-state stability property", Eur. J. Contr., 1, pp. 24-36, 1995.
33. Special Issue, "Chaos synchronization and control: theory and applications", IEEE Trans. Circ. Syst. I, 44, pp. 853-1039, 1997.
34. J.M.T. Thompson and H.B. Stewart, Nonlinear Dynamics and Chaos; Geometrical Methods for Engineers and Scientists, New York: John Wiley and Sons Ltd., 1986.
35. D.H. van Campen, R.H.B. Fey, F.P.H. van Liempt and A. de Kraker, "Steady-state behaviour of a solar array system with elastic stops", in Proc. IUTAM Symp. on New Applications of Nonlinear and Chaotic Dynamics in Mechanics, Dordrecht: Kluwer, 1997, pp. 303-312.
36. N. van de Wouw, "Steady state behaviour of stochastically excited nonlinear dynamic systems". PhD thesis, Eindhoven University of Technology (1999).
37. N. van de Wouw, H. Nijmeijer and D. van Campen, "Approximation of stochastic non-linear dynamics. A Volterra series approach", Nonlinear Dynamics, 27, pp. 397-409, 2002.
38. R. van den Berg, A. Pogromsky, G. Leonov and J. Rooda, "Design of convergent switched systems", in Lecture Notes in Control and Information Sciences, K. Pettersen, J. Gravdahl and H. Nijmeijer, Eds, Vol. 336, Berlin: Springer, 2006, pp. 291-311.
39. E. van den Eijnde and J. Schoukens, "Steady-state analysis of a periodically excited nonlinear system", IEEE Trans. Circ. Syst., 31, pp. 232-242, 1990.
40. B. van der Pol and B. van der Mark, "Frequency demultiplication", Nature, 120, pp. 363-364, 1927.
41. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed., New York: Springer, 2003.
42. V.A. Yakubovich, "Matrix inequalities method in stability theory for nonlinear control systems: I. Absolute stability of forced vibrations", Autom. Remote Contr., 7, pp. 905-917, 1964.
43. V.A. Yakubovich, G.A. Leonov and A.Kh. Gelig, Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities, Singapore: World Scientific, 2004.
44. T. Yoshizawa, Stability Theory by Liapunov's Second Method, Tokio: The Mathematical Society of Japan, 1966.