**1. Introduction**

26 Nonlinearity, Bifurcation and Chaos – Theory and Applications

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

> 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. However, performance of PWM controllers is affected by delays.

> 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 [5].

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

> 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 reported widely in Digital-PWM switched converters.

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

#### 2 Nonlinearity, Bifurcation and Chaos - Theory and Applications 28 Nonlinearity, Bifurcation and Chaos – Theory and Applications Floquet Exponents and Bifurcations in Switched Converters <sup>3</sup>

Stability, bifurcations and transient response of switched converters with *Delayed PWM controllers* can be studied more efficiently using an analysis of disturbances based on *Floquet theory*. We show that this procedure can be generalized to compute Floquet exponents for any number of delays (*τ*) in the control law (*dk*, so-called *duty cycle*). We compare this approach with other methods which determine stability in switched converters. One of them is the computation of characteristic multipliers based on the jacobian matrix. Another one is the computation of Lyapunov exponents using a numeric routine. Each method gives equivalent information. However Floquet approach is the most appropriated when delays appear since this method does not require the evaluation of the jacobian matrix (its dimension increases when the number of delays is higher). The other two methods have this disadvantage [13].

The chapter is organized as follows. In *Section 2* we present the general procedure to compute Floquet exponents in PWM switched converters. The particular case of a buck converter controlled with digital-PWM controller based on ZAD, FPIC and DELAY schemes, is presented in *Section 3*. The stability of fixed points in delayed PWM switched converters is discussed in *Section 4*, while fold, flip and Neimark-Sacker bifurcations are presented in *Section 5*. Finally, the conclusions and future work are presented in *Section 6*.
