*5.1.1. Control subspaces*

The flip bifurcations have been detected in the following (*ks* − *N*) plane: Subspace I when *τ* = 0. The flip zone in the plane *τ* = 0 is presented in figure 8 (*b*).

**6. Conclusions and future work**

**Author details**

**7. References**

John Alexander Taborda

Fabiola Angulo and Gerard Olivar

*Catalonia (in Spanish)*, 2004.

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In this chapter, we have presented a generalized procedure to compute Floquet exponents for any number of delays (*τ*) in the control law of PWM switched converters. We have investigated the incidence of *delays* in a digital-PWM controller based on two novel techniques: *Zero Average Dynamics* (ZAD) and *Fixed-Point Inducting Control* (FPIC). Principles of *Floquet theory* and *smooth bifurcation theory* were used to define stability regions. The incidence of control parameters in transient response of PWM switched converters will be

Floquet Exponents and Bifurcations in Switched Converters 43

*Universidad del Magdalena - Facultad de Ingeniería - Programa de Ingeniería Electrónica - Magma*

*Universidad Nacional de Colombia - Sede Manizales - Facultad de Ingeniería y Arquitectura - Departamento de Ingeniería Eléctrica, Electrónica y Computación - Percepción y Control Inteligente -*

[1] Angulo F., "Dynamical analysis of PWM-controlled power electronic converters based on the zero average dynamics (ZAD) strategy", *Ph. D. Thesis. Technical University of*

[2] Angulo F., Fossas E. & Olivar G. "Transition From Periodicity to Chaos in a PWM-Controlled Buck Converter with ZAD strategy", *International Journal of*

[3] Angulo F., Ocampo C., Olivar G. & Ramos R., *Nonlinear and Nonsmooth Dynamics in a DC-DC Buck Convreter: Two experimental set-ups* Nonlinear Dynamics, Vol. 46, No. 3, pp.

[4] Angulo F., Olivar G. & Taborda J.A., "Continuation of periodic orbits in a ZAD-strategy

[5] Angulo F., Hoyos F.E. & Taborda J.A., "Principios de la Estrategia de Control Zero Average Dynamics (ZAD)" (in Spanish). Lambert Academic Publishing, Madrid,

[6] Awrejcewicz J., "Bifurcation and Chaos in Simple Dynamical Systems". World Scientific,

[7] Awrejcewicz J., "Three routes to chaos in simple sinusoidally driven oscillators", Journal

[8] Awrejcewicz J., "Numerical analysis of the oscillations of human vocal cords",

[9] Awrejcewicz J., Kudra G. & Lamarque C-H., "Investigation of triple pendulum with impacts using fundamental solution matrices", International Journal of Bifurcation and

controlled buck converter", *Chaos, Solitons and Fractals*, 38, 348–363, 2008.

of Applied Mathematics and Mechanics ZAMM, 71 (2), 1991, 71-79.

analyzed in a future work using a similar Floquet-based procedure.

*Ingeniería - Santa Marta D.T.C.H., 2121630, Colombia*

*Bloque Q, Campus La Nubia, Manizales, 170003 - Colombia*

*Bifurcations and Chaos*, 15, 10, 3245–3264, 2005.
