**9. References**

	- [7] He JH (2008) Recent developments of the homotopy perturbation method. Top. Meth. Nonlin. Anal*.*, 31: 205–209.

**Chapter 0**

**Chapter 2**

**Floquet Exponents and Bifurcations**

John Alexander Taborda, Fabiola Angulo and Gerard Olivar

(*PWM*) is the most used method to control power converters [11]-[12].

However, performance of PWM controllers is affected by delays.

reported widely in Digital-PWM switched converters.

cited.

Switched power converters are finding wide applications in the area of electrical energy conditioning. Many electronic devices have power converters to achieve high conversion efficiency and therefore low heat waste. Some of them are: drivers for industrial motion control, battery chargers, uninterruptible power supplies (UPS), electric vehicles, laptops, gadgets and mobile phones. Therefore control of power converters in order to optimize conversion efficiency is a current and challenging research topic. Pulsewidth modulation

Digital-PWM controllers are a novel alternative to control power converters. These controllers have many advantages as programmability, high flexibility, reliability and easy implementation of advanced control algorithms. They can be designed with *delays* in the measured variables in order to guarantee the necessary computing time of the signal control.

In this chapter, we investigate the incidence of *delays* in a digital-PWM controller based on two novel techniques: *Zero Average Dynamics* (ZAD) and *Fixed-Point Inducting Control* (FPIC). Both control strategies have been developed, applied and widely analyzed in the last decade

*Floquet theory* and *smooth bifurcation theory* can be used to define stability regions and to find optimum parameter sets (see for example [6]-[9]). In our case, three parameters should be tuned in the digital-PWM controller. Each parameter is denoted as: *ks* in ZAD strategy, *N* in

The 3D-parameter space (*ks*,*N*,*τ*) of the delayed PWM controller is analyzed and stability regions are bounded by *Flip*, *Fold* and *Neimark-Sacker* transitions. The presence of the three smooth bifurcations in the same nonlinear circuit is not common and this fact has not been

and reproduction in any medium, provided the original work is properly cited.

©2012 Angulo et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

FPIC technique and *τ* is the number of delay periods in the measured variables [1].

Additional information is available at the end of the chapter

**in Switched Converters**

http://dx.doi.org/10.5772/51330

**1. Introduction**

[5].

