Piecewise-linear identification of nonlinear dynamical systems in view of their circuit implementations

IEEE Transactions on Circuits and Systems I: Regular Papers

Volumn 54, Issue 7, 2007, Pages 1542-1554

  De Feo, Oscar ^a   Storace, Marco ^b  

^a UNIVERSITY COLLEGE CORK   (Ireland)

^b UNIVERSITY OF GENOA   (Italy)

Author keywords

Bifurcation; Function approximation; Identification; Modeling; Nonlinear systems; Piecewise linear (PWL) approximation techniques

Indexed keywords

APPROXIMATION THEORY; BIFURCATION (MATHEMATICS); IDENTIFICATION (CONTROL SYSTEMS); NONLINEAR CONTROL SYSTEMS; PIECEWISE LINEAR TECHNIQUES; PRINCIPAL COMPONENT ANALYSIS;

CONTROL PARAMETER; NONLINEAR DYNAMICAL SYSTEM;

ELECTRIC NETWORK ANALYSIS;

EID: 34547125805     PISSN: 10577122     EISSN: None     Source Type: Journal    
DOI: 10.1109/TCSI.2007.899613     Document Type: Article

References (67)

1. S. Strogatz, "Exploring complex networks," Nature, vol. 410, pp. 268-276, Mar. 2001.
2. G. D'anna, P. Mayor, A. Barrat, V. Loreto, and F. Nori, "Observing brownian motion in vibration-fluidized granular matter," Nature, vol. 424, pp. 909-912, Aug. 2003.
3. W. Maass, T. Natschläger, and H. Markram, "Real-time computing without stable states: A new framework for neural computation based on perturbations," Neural Comput., vol. 14, no. 11, pp. 2531-2560, Nov. 2002.
4. T. Mestl, C. Lemay, and L. Glass, "Chaos in high dimensional neural and gene networks," Phys. D, vol. 98, no. 1, pp. 33-52, Nov. 1996.
5. J. Eckmann and E. Moses, "Curvature of co-links uncovers hidden thematic layers in the world-wide web," Proc. Nat. Acad. Sci., vol. 99, no. 9, pp. 5825-5829, Apr. 2002.
6. J. D. W. Tschacher, The Dynamical Systems Approach to Cognition - Concepts and Empirical Paradigms Based on Self-Organization, Embodiment, and Coordination Dynamics. Singapore: World Scientific, 2003.
7. R. Ferrer i Cancho and R. Solé, "The small world of human language," Proc. Royal Soc. London B, vol. 268, no. 1482, pp. 2261-2265, Nov. 2001.
8. W. B. Arthur, "Complexity and the economy," Science, vol. 284, no. 5411, pp. 107-109, Apr. 1999.
9. J. Montoya and R. Solé, "Small world patterns in food webs," J. Theor. Biol., vol. 214, no. 3, pp. 405-412, Feb. 2002.
10. H. Jeong, B. Tombor, R. Albert, Z. Oltavi, and A. Barabási, "The large-scale organization of metabolic networks," Nature, vol. 407, no. 6804, pp. 651-654, Oct. 2000.
11. B. A. Carreras, V. E. Lynch, I. Dobson, and D. E. Newman, "Critical points and transitions in an electric power transmission model for cascading failure blackouts," Chaos, vol. 12, no. 4, pp. 985-994, Dec. 2002.
12. R. F. i Cancho, C. Janssen, and R. Solé, "Topology of technology graphs: Small world patterns in electronic circuits," Phys. Rev. E vol. 64, no. 4, p. 046 119(1-5), Oct. 2001.
13. A. Visser and K. Nagel, "Traffic simulation," Future Generation Comput. Syst., vol. 17, no. 5, pp. 625-626, Mar. 2001.
14. L. Chua, CNN: A Paradigm for Complexity, ser. A. Singapore: World Scientific, 1998, vol. 31.
15. P. Julián, A. Desages, and O. Agamennoni, "High-level canonical piecewise-linear representation using a simplicial partition," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 4, pp. 463-480, Apr. 1999.
16. P. Julián, A. Desages, and B. D'Amico, "Orthonormal high level canonical PWL functions with applications to model reduction," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 5, pp. 702-712, May 2000.
17. M. Storace, P. Julián, and M. Parodi, "Synthesis of nonlinear multiport resistors: A PWL approach," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 8, pp. 1138-1149, Aug. 2002.
18. M. Storace, L. Repetto, and M. Parodi, "A method for the approximate synthesis of cellular nonlinear networks - Part 1: Circuit definition," Int. J. Circuit Theory Appl., vol. 31, no. 3, pp. 277-297, May-Jun. 2003.
19. M. Storace and M. Parodi, "Towards analog implementations of PWL two-dimensional nonlinear functions," Int. J. Circuit Theory Appl. vol. 33, no. 2, pp. 147-160, Mar.-Apr. 2005.
20. M. Parodi, M. Storace, and P. Julián, "Synthesis of multiport resistors with piecewise-linear characteristics:AMixed-signal architecture," Int. J. Circuit Theory Appl., vol. 33, no. 4, pp. 307-319, Jul.-Aug. 2005.
21. M. Storace and O. De Feo, "Piecewise-linear approximation of nonlinear dynamical systems," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 4, pp. 830-842, Apr. 2004.
22. M. Storace and O. De Feo, "PWL approximation of nonlinear dynamical systems_Part I: Structural stability," J. Phys., vol. 22, pp. 208-221, 2005.
23. L. Repetto, M. Storace, and M. Parodi, "A method for the approximate synthesis of cellular nonlinear networks - Part 2: Circuit reduction," Int. J. Circuit Theory Appl., vol. 31, no. 3, pp. 299-313, May-June 2003.
24. A. Hodgkin and A. Huxley, "A quantitative description of membrane current and its applications to conduction in nerve," J. Physiol. vol. 117, pp. 500-544, 1952.
25. M. Parodi and M. Storace, "On a circuit representation of the hodgkin and huxley nerve axon membrane equations," Int. J. Circuit Theory Appl., vol. 25, no. 2, pp. 115-124, Mar.-Apr. 1997.
26. Y. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed. New York: Springer-Verlag, 1998.
27. M. Gaggero, M. Parodi, and M. Storace, "Multiresolution PWL approximations," presented at the Proc. Eur. Conf. Circuit Theory and Design (ECCTD), Cork, Ireland, Aug. 29-Sep. 2 2005, paper 051, unpublished.
28. O. De Feo and M. Storace, "PWL approximation of nonlinear dynamical systems Part - II: Identification issues," J. Phys., vol. 22, pp. 30-42, 2005.
29. O. De Feo and M. Hasler, Modelling Diversity by Chaos and Classification by Synchronization. Dordrecht, Germany: Kluwer, 2003, pp. 25-40.
30. O. De Feo, "Self-emergence of chaos in the identification of irregular periodic behavior," Chaos, vol. 13, no. 4, Dec. 2003.
31. O. De Feo, "Qualitative resonance of Shil'nikov-like strange attractors - Part I: Experimental evidence," Int. J. Bifurc. Chaos, vol. 14, no. 3, pp. 873-891, Mar. 2004.
32. O. De Feo, "Qualitative resonance of Shil'nikov-like strange attractors - Part II: Mathematical analysis," Int. J. Bifurc. Chaos, vol. 14, no. 3, pp. 893-912, Mar. 2004.
33. G. Yule, "On a method for investigating periodicites in disturbed series with special reference to Wolfe's sunspot number," Phil. Trans. Roy. Soc., vol. 226, pp. 267-298, 1927.
34. N. Packard, J. Crutchfield, J. Farmer, and R. Shaw, "Geometry from a time series," Phys. Rev. Lett., vol. 45, no. 9, pp. 712-716, Sep. 1980.
35. D. Ruelle, Chaotic Evolution and Strange Attractors: The Statistical Analysis of Time Series for Deterministic Nonlinear Systems. Cambridge, MA: Cambridge Univ. Press, 1989.
36. F. Takens, Detecting Strange Attractors in Fluid Turbolence, ser. Lecture notes in mathematics. Berlin, Germany: Springer, 1981, pp. 366-381.
37. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, 2nd ed. Cambridge, MA: Cambridge University Press, 2003.
38. P. Hegger, H. Kantz, and T. Schreiber, "Practical implementation of nonlinear time series methods: The TISEAN package," Chaos, vol. 9, no. 2, pp. 413-435, Jun. 1999.
39. H. Abarbanel, R. Brown, J. Sidorowich, and L. Tsimring, "The analysis of observed chaotic data in physical systems," Rev. Modern Phys., vol. 65, no. 4, pp. 1331-1392, Oct. 1993.
40. J. Gibson, J. Farmer, M. Casdagli, and S. Eubank, "An analytic approach to practical state-space reconstruction," Phys. D, vol. 57, no. 1-2, pp. 1-30, Jun. 1992.
41. M. Casdagli, S. Eubank, J. Farmer, and J. Gibson, "State-space reconstruction in the presence of noise," Phys. D, vol. 51, no. 1-3, pp. 52-98, Aug. 1991.
42. P. van Overschee and B. De Moor, Subspace Identification of Linear Systems: Theory, Implementation, Applications. Dordrecht, The Netherlands: Kluwer Academic, 1996.
43. R. Mañé, On the Dimension of the Compact Invariant Set of Certain Non-Linear Maps, ser. Lecture notes in mathematics. Berlin, Germany: Springer, 1981, pp. 230-242.
44. M. Small, Applied Nonlinear Time Series Analysis: Applications in Physics, Physiology and Finance. Singapore: World Scientific, 2005.
45. H. Hotelling, "Analysis of a complex of statistical variables into principal components," J. Educ. Psychol., vol. 24, pp. 417-441, 1933, 498-520.
46. K. Karhunen, "Uber lineare methoden in der wahrscheinlichkeitsrechnung," Amer. Acad. Sci., Fennicade, Ser. A, I vol. 37, pp. 3-79, 1947, translation: RAND Corporation, Santa Monica, California, Rep. T-131, Aug. 1960.
47. M. Loeve, Fonctions aléatories de second ordre. Paris: Herman, 1948.
48. A. Newman, Model Reduction via the Karhunen-Loeve Expansion Part I: An Exposition Institute for System Research, University of Maryland, Tech. Rep. 96-32, 1996.
49. A. Newman, Model Reduction via the Karhunen-Loeve Expansion Part I: Some Elementary Examples Institute for System Research, University of Maryland, Tech. Rep. 96-32, 1996.
50. L. Cao, "Practical method for determining the minimum embedding dimension of a scalar time series," Phys. D, vol. 110, no. 1-2, pp. 43-50, Dec. 1997.
51. O. De Feo, "Transforming subharmonic chaos to homoclinic chaos suitable for pattern recognition," Int. J. Bifurc. Chaos, vol. 15, no. 10, pp. 3345-3357, Oct. 2005.
52. O. De Feo, "Tuning chaos synchronization and anti-synchronization for applications in temporal pattern recognition," Int. J. Bifurcation Chaos, vol. 15, no. 12, pp. 3905-3921, Dec. 2005.
53. M. Toledo-Rodriguez, B. Blumenfeld, C. Wu, J. Luo, B. Attali, P. Goodman, and H. Markram, "Correlation maps allow neuronal electrical properties to be predicted from single-cell gene expression profiles in rat neocortex," Cerebral Cortex, vol. 14, no. 12, pp. 1310-1327, Dec. 2004.
54. G. Maggio, O. De Feo, and M. Kennedy, "Nonlinear analysis of the Colpitts oscillator and applications to design," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 9, pp. 1118-1130, Sep. 1999.
55. O. De Feo, G. Maggio, and M. Kennedy, "The Colpitts oscillator: Families of periodic solutions and their bifurcations," Int. J. Bifurc. Chaos, vol. 10, no. 5, pp. 935-958, May 2000.
56. M. Candaten and S. Rinaldi, "Peak-to-peak dynamics: A critical survey," Int. J. Bifurc. Chaos, vol. 10, no. 8, pp. 1805-1819, Aug. 2000.
57. H. Markram, J. Lubke, M. Frotscher, A. Roth, and B. Sakmann, "Physiology and anatomy of synaptic connections between thick tufted pyramidal neurones in the developing rat neocortex," J. Physiol., vol. 500, no. 2, pp. 409-440, Apr. 1997.
58. A. Gupta, Y. Wang, and H. Markram, "Organizing principles for a diversity of GABAergic interneurons and synapses in the neocortex," Science, vol. 287, no. 5451, pp. 273-278, Jan. 2000.
59. Y. Wang, A. Gupta, M. Toledo-Rodriguez, C. Wu, and H. Markram, "Anatomical, physiological, molecular and circuit properties of nest basket cells in the developing somatosensory cortex," Cerebral Cortex, vol. 12, no. 4, pp. 395-410, Apr. 2002.
60. Online. Available: Http://bluebrainproject.epfl.ch/
61. M. Orr, Introduction to Radial Basis Function Networks intro.ps [Online]. Available: Http://www.anc.ed.ac.uk/mjo/
62. M. Orr, Recent Advances in Radial Basis Function Networks recad.ps [Online]. Available: Http://www.anc.ed.ac.uk/mjo/
63. L. Theogarajan and L. Akers, "A scalable low-voltage analog Gaussian radial basis circuit," IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 44, no. 11, pp. 977-979, Nov. 1997.
64. J. Madrenas, M. Verleysen, P. Thissen, and J. Voz, "A CMOS analog circuit for Gaussian functions," IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 43, no. 1, pp. 70-74, Jan. 1996.
65. L. Repetto, "A Piecewise-Linear Approach to Approximate Circuit Syntheses of Nonlinear Systems" Ph.D. dissertation, Biophysical and Electronic Engineering Department, University of Genoa, Genova, Italy, 2004 Online. Available: Http://www.dibe-mail.dibe.unige.it/department/ ncas/REPETTO/Thesis_C.pdf
66. L. Repetto, M. Parodi, and M. Storace, "A procedure for the computation of accurate PWL approximations of nonlinear dynamical systems," Int. J. Circuit Theory Appl., vol. 34, no. 2, pp. 237-248, Mar.-Apr. 2006.
67. M. Storace and F. Bizzarri, "Towards accurate PWL approximations of parameter-dependent nonlinear dynamical systems with equilibria and limit cycles," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 54, no. 3, pp. 620-631, Mar. 200