Bifurcations in a Generalization of the ZAD Technique: Application to a DC-DC Buck Power Converter

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2012, Article ID 520296, 13 pages doi:10.1155/2012/520296 Research Article Bifurcations in a Generalization of the ZAD Technique: Application to a DC-DC Buck Power Converter Ludwing Torres,1 Gerard Olivar,1 and Simeón Casanova2 1 Percepción y Control Inteligente, Departamento de Ingenierı́a Eléctrica, Facultad de Ingenierı́a y Arquitectura, Universidad Nacional de Colombia, Sede Manizales, Electrónica y Computación, Bloque Q, Campus La Nubia, Manizales, Colombia 2 ABC Dynamics, Departamento de Matemáticas y Estadı́stica, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Colombia, Sede Manizales, Bloque Y, Campus La Nubia, Manizales, Colombia Correspondence should be addressed to Gerard Olivar, Received 21 March 2012; Accepted 30 April 2012 Academic Editor: Ahmad M. Harb Copyright q 2012 Ludwing Torres et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A variation of ZAD technique is proposed, which is to extend the range of zero averaging of the switching surface in the classic ZAD it is taken in a sampling period, to a number K of sampling periods. This has led to a technique that has been named K-ZAD. Assuming a specific value for K  2, we have studied the 2-ZAD technique. The latter has presented better results in terms of stability, regarding the original ZAD technique. These results can be demonstrated in different state space graphs and bifurcation diagrams, which have been calculated based on the analysis done about the behavior of this new strategy. 1. Introduction Currently, power electronics has an important place in industry. This is largely due to the very extensive number of applications derived from these systems, including control of power converters. This was the main motivation for researchers worldwide to develop new advances in this field and has promoted the investigation on many mathematical models. They correspond usually to variable structure systems, chaos, and control. The practical goal is to obtain better devices or new and improved mechanisms for use in electronic controllers. Bifurcation, chaos, and control in electronic circuits have been reported in many papers such as 1, 2. Regarding power electronics, early works can be found in the literature from the 1980s when first observed in 3, 4. In 1989 several authors such as Krein and Bass 5, and 2 Mathematical Problems in Engineering Deane and Hamill 6 began to study chaos in various power electronic circuits. In his work Wood 7 contributed to the understanding of chaos, showing in phase diagrams that the pattern of the trajectories was messy as small variation on initial conditions and parameters in the system were performed. Meanwhile Deane and Hamill reported some of the best works on bifurcations and chaos applied to electronic circuits. Their studies were based on computer simulations and laboratory experiments 3, 4, 6. Because of the multiple side effects that generation of chaos caused in these systems, several authors proceeded to develop different control techniques, such as 8 by Ott et al., who found a way of controlling unstable orbits coexisting with chaos. They used small disturbances. This resulted in the so well-known OGY method after the names of the authors Ott, Grebogi, and Yorke. Pyragas 9 contributed to this topic with a feedback scheme using a time-delayed control known as time-delayed autosynchronization TDAS. Also the work by Utkin is very important in the control literature. He studied pulsewidth modulation PWM systems and variable structure systems 10. He introduced a now very popular control technique so-called sliding-mode control SMC and many applications were proposed 11. These studies were taken later as a starting point by Carpita et al. 12. In his work, Carpita presents a sliding mode controller for a uninterrupted power system UPS. Carpita uses a sliding surface which is a linear combination of error variable and its derivative. Some time after, a new technique, both conceptually different from sliding-mode control although very related to it and as a practical implementation to SMC, appeared. It was called zero average dynamics ZAD by the authors. This control technique forces the system to have, at each clock cycle, a zero average of the sliding surface instead to be zero all the time, as in sliding-mode control. Application of this technique to a buck converter with centered pulse and an approximation scheme needed for the practical implementations 12 has obtained very good performance. Robustness, low stationary error, and fixed switching frequency have also been achieved 13. It has been observed that as the main parameter in the surface ks is decreased, chaotic behavior appears which is not desirable in practice. Then, several additional techniques such as TDAS mentioned before or fixed point induced control FPIC must be implemented in order to guarantee a wider operation range 13. FPIC has obtained the best results with regards to stabilization and chaos control. Bifurcation diagrams have shown flip or period-doubling bifurcations followed by border-collision bifurcations due to the saturation of the limit cycle. This sequence of bifurcations leads the system to chaotic operation for low values of ks . When saturation of the limit cycle appears, the ZAD technique is degraded and thus zero average in each cycle is lost. In order to solve this problem, we generalize the ZAD technique from now on called classical ZAD to the so-called K-ZAD technique. This generalization allows the surface to be of zero average not in every cycle as classical ZAD, but in a sequence of K cycles. Being a weaker condition one can choose the duty cycles in such a way that they are not saturated at least not so frequently and thus zero average and stabilization in a periodic orbit is obtained. Specifically, our study has focused on the value K  2, which gives rise to 2-ZAD, that is, two sampling periods for the zero average in the surface. We compare this generalization with the classical ZAD through bifurcation diagrams. The remaining of this paper is organized as follows. In Section 2 we explain in detail mathematical modelling of this technique and we perform the corresponding algebraic computations. In Section 3 the numerical implementation and the results are discussed. Finally, conclusions and future work are stated in Section 4. Mathematical Problems in Engineering E (1) C1 + 3 L (2) C2 C − (2) C3 + R − (1) C4 Figure 1: Scheme of the Buck converter. 2. K-ZAD Strategy The scheme for the buck converter is shown in Figure 1. We donot take into account the diode full dynamics and thus we assume always continuous conduction mode alt (...truncated)