study of bifurcation and chaos in dc-dc boost converter using discrete-time map
TRANSCRIPT
Study of Bifurcation and Chaos in DC-DC Boost
Converter using Discrete-Time Map
Sajid Iqbal Ph.D student
Harbin Institute of Technology
• Zheng Jie
Contents
• Chaos and bifurcation
• DC-DC boost converter
• Discrete-time map
• Tools for finding chaotic dynamics
o Time series
o Bifurcation diagram
‘Deterministic Chaos’ is an unstable aperiodic behavior in deterministic dynamical system, which shows sensitive dependence on initial conditions.
Edward Lorenz coined the term ‘Butterfly Effect’.
‘Bifurcation’ is a sudden qualitative change in the behavior of a dynamical system caused by the variation of its parameters.
In a boost chopper, the output voltage is always more or equal to the input voltage.
DC-DC converters have many sources of nonlinearity like
Semiconductors devices: diodes, transistors
Passive components: capacitors, inductors and
Feedback control circuits
A ‘time series plot’ is a display of data points that shows how values have changed over uniform time intervals
A ‘bifurcation diagram’ is a visual summary chart of the behaviors exhibited by a dynamical system, when some parameters are varied.
For developing a discrete-time system, the dynamical variables of the continuous-time system are sampled once every period.
The goal is to create a mapping, difference equation or iterative map as
Where ‘Xn’ is the state vector at the nth switching instant and k represents the feedback factor.
1 n(x ,k)nx f
The discrete-time model of Tse is revisited. As an first-order iterative map
‘vn’ is the voltage and ‘dn’ is the duty cycle during the nth period.
1 n( ,d )n nv f v
The iterates settle down to a fixed value
Period-1 orbit
1n nv v
The iterated solutions reappear every second value
Period-2 orbit
2n nv v
Time series plots of boost converter (cont.)
Aperiodic behavior
Bifurcation diagram for boost converter.
Don't curse the darkness, light a candle. Yǔqí yǔ zhōu hēi àn, bùrú rén qí lā zhū.