# American Institute of Mathematical Sciences

2017, 22(2): 569-584. doi: 10.3934/dcdsb.2017027

## Randomly perturbed switching dynamics of a dc/dc converter

 Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382355, India

Received  January 2016 Revised  May 2016 Published  December 2016

Fund Project: The author acknowledges research support from DST SERB Project No. EMR/2015/000904

In this paper, we study the effect of small Brownian noise on a switching dynamical system which models a first-order DC/DC buck converter. The state vector of this system comprises a continuous component whose dynamics switch, based on the ON/OFF configuration of the circuit, between two ordinary differential equations (ODE), and a discrete component which keeps track of the ON/OFF configurations. Assuming that the parameters and initial conditions of the unperturbed system have been tuned to yield a stable periodic orbit, we study the stochastic dynamics of this system when the forcing input in the ON state is subject to small white noise fluctuations of size $\varepsilon$, $0<\varepsilon \ll 1$. For the ensuing stochastic system whose dynamics switch at random times between a small noise stochastic differential equation (SDE) and an ODE, we prove a functional law of large numbers which states that in the limit of vanishing noise, the stochastic system converges to the underlying deterministic one on time horizons of order $\mathscr{O}(1/\varepsilon ^ν)$, $0 ≤ ν < 2/3$.

Citation: Chetan D. Pahlajani. Randomly perturbed switching dynamics of a dc/dc converter. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 569-584. doi: 10.3934/dcdsb.2017027
##### References:
 [1] S. Banerjee and K. Chakrabarty, Nonlinear modeling and bifurcations in the boost converter, IEEE Transactions on Power Electronics, 13 (1998), 252-260. [2] S. Banerjee, M. S. Karthik, G. Yuan and J. A. Yorke, Bifurcations in one-dimensional piecewise smooth maps-Theory and applications in switching circuits, IEEE Transactions on Circuits and Systems-Ⅰ: Fundamental Theory and Applications, 47 (2000), 389-394. doi: 10.1109/81.841921. [3] S. Banerjee and G. C. Verghese (editors), Nonlinear Phenomena in Power Electronics Wiley, 2001. [4] G. K. Basak, A. Bisi and M. K. Ghosh, Stability of degenerate diffusions with state-dependent switching, Journal Math. Anal. Appl., 240 (1999), 219-248. doi: 10.1006/jmaa.1999.6610. [5] P. Billingsley, Convergence of Probability Measures, second edition, John Wiley & Sons Inc., 1999. doi: 10.1002/9780470316962. [6] D. Chatterjee and D. Liberzon, On stability of randomly switched nonlinear systems, IEEE Transactions on Automatic Control, 52 (2007), 2390-2394. doi: 10.1109/TAC.2007.904253. [7] D. Chatterjee and D. Liberzon, Stabilizing randomly switched systems, SIAM Journal on Control and Optimization, 49 (2011), 2008-2031. doi: 10.1137/080726720. [8] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Theory and Applications, Springer, 2008. [9] M. di Bernardo, F. Garofalo, L. Glielmo and F. Vasca, Switchings, bifurcations and chaos in DC/DC converters, IEEE Transactions on Circuits and Systems-Ⅰ: Fundamental Theory and Applications, 45 (1998), 133-141. [10] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, second edition, Springer, 1998. doi: 10.1007/978-1-4612-5320-4. [11] S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons Inc., New York, 1986. doi: 10.1002/9780470316658. [12] E. Fossas and G. Olivar, Study of chaos in the buck converter, IEEE Transactions on Circuits and Systems-Ⅰ: Fundamental Theory and Applications, 43 (1996), 13-25. [13] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Third Edition, Springer, 2012. doi: 10.1007/978-3-642-25847-3. [14] M. Hasler, V. Belykh and I. Belykh, Dynamics of stochastically blinking systems, Part Ⅰ: Finite time properties, SIAM Journal on Applied Dynamical Systems, 12 (2013), 1007-1030. doi: 10.1137/120893409. [15] M. Hasler, V. Belykh and I. Belykh, Dynamics of stochastically blinking systems, Part Ⅱ: Asymptotic properties, SIAM Journal on Applied Dynamical Systems, 12 (2013), 1031-1084. doi: 10.1137/120893410. [16] D. C. Hamill, J. H. B. Deane and D. J. Jeffries, Modeling of chaotic DC-DC converters by iterated nonlinear mappings, IEEE Trans. Power Electron., 7 (1992), 25-36. [17] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Springer-Verlag, 1991. doi: 10.1007/978-1-4612-0949-2. [18] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, Journal of Mathematical Analysis and Applications, 334 (2007), 69-84. doi: 10.1016/j.jmaa.2006.12.032. [19] A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, J. Sound and Vibration, 145 (1991), 279-297. [20] G. A. Pavliotis and A. M. Stuart, Multiscale Methods, Averaging and Homogenization, Texts in Applied Mathematics, 53. Springer, New York, 2008. [21] S. W. Shaw and P. J. Holmes, A periodically forced piecewise linear oscillator, J. Sound and Vibration, 90 (1983), 129-155. doi: 10.1016/0022-460X(83)90407-8. [22] D. J. W. Simpson and R. Kuske, Stochastically perturbed sliding motion in piecewise-smooth systems, Discrete Cont. Dyn. Syst. Ser. B, 19 (2014), 2889-2913. doi: 10.3934/dcdsb.2014.19.2889. [23] D. J. W. Simpson and R. Kuske, The positive occupation time of Brownian motion with two-valued drift and asymptotic dynamics of sliding motion with noise, Stoch. Dyn. , 14 (2014), 1450010, 23pp. [24] D. J. W. Simpson and R. Kuske, Stochastic perturbations of periodic orbits with sliding, J. Nonlin. Sci., 25 (2015), 967-1014. doi: 10.1007/s00332-015-9248-7. [25] G. Yin and C. Zhu, Properties of solutions of stochastic differential equations with continuous-state-dependent switching, Journal of Differential Equations, 249 (2010), 2409-2439. doi: 10.1016/j.jde.2010.08.008. [26] G. Yin and C. Zhu, Hybrid Switching Diffusions. Properties and Applications, Springer, New York, . doi: 10.1007/978-1-4419-1105-6.

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##### References:
 [1] S. Banerjee and K. Chakrabarty, Nonlinear modeling and bifurcations in the boost converter, IEEE Transactions on Power Electronics, 13 (1998), 252-260. [2] S. Banerjee, M. S. Karthik, G. Yuan and J. A. Yorke, Bifurcations in one-dimensional piecewise smooth maps-Theory and applications in switching circuits, IEEE Transactions on Circuits and Systems-Ⅰ: Fundamental Theory and Applications, 47 (2000), 389-394. doi: 10.1109/81.841921. [3] S. Banerjee and G. C. Verghese (editors), Nonlinear Phenomena in Power Electronics Wiley, 2001. [4] G. K. Basak, A. Bisi and M. K. Ghosh, Stability of degenerate diffusions with state-dependent switching, Journal Math. Anal. Appl., 240 (1999), 219-248. doi: 10.1006/jmaa.1999.6610. [5] P. Billingsley, Convergence of Probability Measures, second edition, John Wiley & Sons Inc., 1999. doi: 10.1002/9780470316962. [6] D. Chatterjee and D. Liberzon, On stability of randomly switched nonlinear systems, IEEE Transactions on Automatic Control, 52 (2007), 2390-2394. doi: 10.1109/TAC.2007.904253. [7] D. Chatterjee and D. Liberzon, Stabilizing randomly switched systems, SIAM Journal on Control and Optimization, 49 (2011), 2008-2031. doi: 10.1137/080726720. [8] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Theory and Applications, Springer, 2008. [9] M. di Bernardo, F. Garofalo, L. Glielmo and F. Vasca, Switchings, bifurcations and chaos in DC/DC converters, IEEE Transactions on Circuits and Systems-Ⅰ: Fundamental Theory and Applications, 45 (1998), 133-141. [10] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, second edition, Springer, 1998. doi: 10.1007/978-1-4612-5320-4. [11] S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons Inc., New York, 1986. doi: 10.1002/9780470316658. [12] E. Fossas and G. Olivar, Study of chaos in the buck converter, IEEE Transactions on Circuits and Systems-Ⅰ: Fundamental Theory and Applications, 43 (1996), 13-25. [13] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Third Edition, Springer, 2012. doi: 10.1007/978-3-642-25847-3. [14] M. Hasler, V. Belykh and I. Belykh, Dynamics of stochastically blinking systems, Part Ⅰ: Finite time properties, SIAM Journal on Applied Dynamical Systems, 12 (2013), 1007-1030. doi: 10.1137/120893409. [15] M. Hasler, V. Belykh and I. Belykh, Dynamics of stochastically blinking systems, Part Ⅱ: Asymptotic properties, SIAM Journal on Applied Dynamical Systems, 12 (2013), 1031-1084. doi: 10.1137/120893410. [16] D. C. Hamill, J. H. B. Deane and D. J. Jeffries, Modeling of chaotic DC-DC converters by iterated nonlinear mappings, IEEE Trans. Power Electron., 7 (1992), 25-36. [17] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Springer-Verlag, 1991. doi: 10.1007/978-1-4612-0949-2. [18] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, Journal of Mathematical Analysis and Applications, 334 (2007), 69-84. doi: 10.1016/j.jmaa.2006.12.032. [19] A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, J. Sound and Vibration, 145 (1991), 279-297. [20] G. A. Pavliotis and A. M. Stuart, Multiscale Methods, Averaging and Homogenization, Texts in Applied Mathematics, 53. Springer, New York, 2008. [21] S. W. Shaw and P. J. Holmes, A periodically forced piecewise linear oscillator, J. Sound and Vibration, 90 (1983), 129-155. doi: 10.1016/0022-460X(83)90407-8. [22] D. J. W. Simpson and R. Kuske, Stochastically perturbed sliding motion in piecewise-smooth systems, Discrete Cont. Dyn. Syst. Ser. B, 19 (2014), 2889-2913. doi: 10.3934/dcdsb.2014.19.2889. [23] D. J. W. Simpson and R. Kuske, The positive occupation time of Brownian motion with two-valued drift and asymptotic dynamics of sliding motion with noise, Stoch. Dyn. , 14 (2014), 1450010, 23pp. [24] D. J. W. Simpson and R. Kuske, Stochastic perturbations of periodic orbits with sliding, J. Nonlin. Sci., 25 (2015), 967-1014. doi: 10.1007/s00332-015-9248-7. [25] G. Yin and C. Zhu, Properties of solutions of stochastic differential equations with continuous-state-dependent switching, Journal of Differential Equations, 249 (2010), 2409-2439. doi: 10.1016/j.jde.2010.08.008. [26] G. Yin and C. Zhu, Hybrid Switching Diffusions. Properties and Applications, Springer, New York, . doi: 10.1007/978-1-4419-1105-6.
The evolution of the components $x(t)$, $y(t)$ of the full state vector $z(t)$ when starting with initial condition $(x_0,1)$ where $x_0 \in (0,x_{\mathsf {ref}})$. Note that $x(t)$ is continuous and piecewise-smooth; it is smooth between corners at the switching times $t_1<s_1<t_2<s_2<\dots$. The function $y(t) \in \{0,1\}$ is piecewise-constant and right-continuous, with jumps at the switching times. The dotted vertical lines at integer times denote the periodic clock pulse which triggers the OFF $\to$ ON transition
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