Coexistence of solutions in a DC-DC Buck converter controlled by sine wave

Abstract

Low power systems are widely used in robotics and industrial areas; therefore, modeling and systems analysis provide reliable and best designs of such systems. In that way, a Buck converter controlled by P W M is investigated using a new design, replacing the T periodic ramp signal by a T periodic sine wave for two reasons: the sine wave is easier to generate and analyze, and in order to get rid of non-linear qualities of the ramp signal. Then, bifurcation diagrams are obtained varying the parameters ascending and descending in the new Buck converter design to find coexisting attractors, which are normally an undesired behavior in nonlinear systems or useful for some applications. Although it is demonstrated that these bifurcation diagrams are not the sufficient remedy to find coexistence of solutions, it is also shown that they are a good tool. Once bifurcation diagrams show the range where coexistence of solutions appear, we proceed to study in that range the shape of the basins and the system solutions, with the aim of determining the specific regions in which the system presents different behaviors. This is because many practical applications today need not only periodic solutions but also more complex ones. Finally, basins of attraction are obtained and studied for some parameters of the system. Bounded and fractal regions are observed; moreover, the evolution of the attractors in the time domain for each parameter value are shown.Index Terms-buck converter, bifurcation diagrams, basins of attraction, fractal regions, attractors.

Bifurcation Phenomena Studies of a Voltage Controlled Buck-Inverter Cascade System

“…In this section, we will study the initial states sensitivity of the Buck-inverter, considering that the system starts from different initial states (as mentioned before, the bifurcation diagrams shown in Figure 4a,c are derived by setting the steady state of the previous parameters as the initial states). References [13,14,19] show that the Buck converter and inverter have coexistence of attractors, and that the attractors are different corresponding to different initial values. As for the Buck-inverter, there are four state variables.…”

“…Particularly, the coexistence of the attractor's phenomenon was found in Buck converters. Starting from different initial values, the converters converged to different steady states [13,14]. Flip bifurcation and chaos were found in a three-state boost switching regulators, and the results show the system exhibited a typical route to chaos via period-doubling [15].…”

Design of PID controllers using Filippov's method for stable operation of DC–DC converters

“…Because (19) contains no terms dependant on any of the controller parameters, we can say that the controller has no effect on the occurrence of saddle-node bifurcations.…”

“…On the other hand, in [18], the authors discuss the use of a feed-forward controller where the amplitude of the ramp signal is proportional to the input voltage. In [19], the authors propose shaping the ramp signal so that it becomes similar to the control signal. The ramp has the capability to change its slope and offset voltage.…”

“…Previous works [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] have suggested that the Jacobian of the Poincaré map or the monodromy matrix can be used to predict these fast-scale instabilities, and hence, it is possible to be used in the design of the controller. The authors in [27] use Filippov's method for the prediction of perioddoubling bifurcations for the buck converter.…”