Switched dynamical systems with double periodic inputs: an analysis tool and its application to the buck-boost converter

This paper considers a class of switched dynamical systems to which two periodic inputs are applicable. We derive a one-dimensional (1-D) return map and its derivative analytically for arbitrary shapes of the inputs. Based on the derivative, we introduce a slope chart that displays the relationship between the slope of the return map and that of the inputs. Applying the slope chart to a simplified model of the buck-boost converter (SBBC) with two periodic triangular inputs, we obtain a sufficient condition for chaos generation and that for the stability of the periodic behavior. The conditions guarantee that the first input can cause chaotic or periodic behavior and that the second input can change from chaotic behavior into periodic behavior and vice versa. A circuit model of the system is also proposed and the periodic and chaotic behaviors are demonstrated in the laboratory.

[1]  T. Tsubone,et al.  Stabilizing and destabilizing control for a piecewise-linear circuit , 1998 .

[2]  H. Torikai,et al.  Chaotic artificial neural system and its control , 1995 .

[3]  Hiroyuki Torikai,et al.  Synchronization of chaos and its itinerancy from a network by occasional linear connection , 1998 .

[4]  Jonathan H. B. Deane,et al.  Chaos in a current-mode controlled boost DC-DC converter , 1992 .

[5]  Krishnendu Chakrabarty,et al.  Control of chaos in DC-DC converters , 1998 .

[6]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[7]  D. C. Hamill.  POWER ELECTRONICS : A FIELD RICH IN NONLINEAR DYNAMICS , 1995 .

[8]  James A. Yorke,et al.  Ergodic transformations from an interval into itself , 1978 .

[9]  T. Saito.  A chaos generator based on a quasi-harmonic oscillator , 1985 .

[10]  A. Sharkovsky,et al.  Chaos in Some 1-D Discontinuous Maps that Apper in the Analysis of Electrical Circuits , 1993 .

[11]  Michael C. Mackey,et al.  Chaos, Fractals, and Noise , 1994 .

[12]  Chern,et al.  Goal-oriented scheme for taming chaos with a weak periodic perturbation: Experiment in a diode resonator. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  J. Keener.  Chaotic behavior in piecewise continuous difference equations , 1980 .

[14]  Yu-Kai Chen,et al.  Modeling PWM DC/DC converters out of basic converter units , 1998 .

[15]  S. Daan,et al.  Timing of human sleep: recovery process gated by a circadian pacemaker. , 1984, The American journal of physiology.

[16]  Hiroyuki Torikai,et al.  Return Map Quantization from an Integrate-and-Fire Model with Two Periodic Inputs , 1999 .

[17]  James A. Yorke,et al.  Border-collision bifurcations in the buck converter , 1998 .

[18]  Goldhirsch,et al.  Taming chaotic dynamics with weak periodic perturbations. , 1991, Physical review letters.

[19]  E. Ott,et al.  Controlling Chaotic Dynamical Systems , 1991, 1991 American Control Conference.

[20]  Chaos and fundamental bifurcation phenomena from a relaxation oscillator with periodic thresholds , 1995 .

[21]  J.H.B. Deane,et al.  Improvement of power supply EMC by chaos , 1996 .

[22]  Y. Lai,et al.  Controlling chaotic dynamical systems , 1997 .

[23]  David C. Hamill,et al.  Subharmonics and chaos in a controlled switched-mode power converter , 1988 .