Abstract
This chapter deals with the analysis of the dynamic behavior of a controlled single-phase H-bridge inverter. The authors show that in addition to border collision bifurcation, when it is controlled with a time-delayed controller or with a dynamic controller that increases the system dimension, the H-bridge inverter can exhibit several other types of behaviors such as Neimark-Sacker bifurcation, quasi-periodicity, and coexistence of different periodic behaviors, as well as coexistence between periodic and chaotic behaviors. The proposed controllers are of different types. In addition to the Fixed-Point Induced Controller (FPIC), the authors also present the Time-Delayed Feedback Controller (TDFC) and the dynamic linear controller, such as the proportional-integral controller. The main issue of this chapter is to perform analysis within and beyond the stability region. Analytic calculation and numerical simulations are presented to confirm the obtained results.
Top1. Introduction
Power electronics is a discipline that has emerged from the need to convert electrical energy. Its field of application is wide and concerns industrial, commercial, residential and also aerospace environments. Nowadays, there is a great interest to use green energy such as the solar and wind energy, there is also a need to use mobile electrical devices supplied by batteries or fuel cells. Knowing that such power sources supply a low DC voltage, then the design of electronic devices that convert the low DC signal into a high AC signal similar to power that would be available at an electrical wall outlet becomes a challenging endeavor. This prescribed aim has been achieved using two stages. The first one is a boosting DC to DC converter and the second is the DC to AC inverter (Luo and Ye, 2004; Rashid, 2001). Today, there are a myriad of such power converters ranging from the most basic to the most complicated and yielding to several optimal behaviors in different senses such as wave forms, power efficiency and circuit simplicity and size.
Power converters are the main devices that helped boosting the use of electronic devices. However, converters have been used long before their behavior is completely understood. Indeed, it’s only on late eighties that Hamill (Hamill and Jefferies, 1988) reported the existence of several nonlinear phenomena in power converters. These include subharmonic oscillations, quasi-periodic operations, bifurcations and chaos.
Chaos in power electronics have intrigued many researchers (Di Bernardo and Tse, 2002; Deane and Hamill, 1990; Tse and Di Bernardo, 2002; Robert and Robert, 2002). In fact, engineers have frequently encountered chaos in power electronics systems, but more often than not this phenomenon was considered as strange and undesirable, hence engineers usually attempted to avoid chaos. During the last two decades, tools of analyzing bifurcations and chaos have been well developed. Therefore, the investigation of the very peculiar aspect of this phenomenon has become an attractive endeavor.
Power converters are basic switching circuits that are modeled by a number of linear differential equations corresponding to different topologies. The switching procedure imposed by different control schemes causes toggling among a set of linear circuits. Therefore the overall dynamics can be easily described by a piecewise linear model. In the case of simple linear models, a discrete map can be obtained by solving and stacking up solutions (Di Bernardo and Vasca, 2000; Robert and El Aroudi, 2006). The discrete-time modeling of the converters has provided an easy way to conceive controllers by using the discrete-time control theory and that can be applied further using the digital pulse width modulator (DPWM) (Feki et al., 2004; Robert et al., 2003; Hamza et al., 2011; Kaoubaâ et al., 2012).
Controlling the converters has many objectives such as ensuring converter closed loop stability and enhancing efficiency and dynamic performances. The switching nature of the continuous time mathematical model or the discrete time model with saturating controller (duty cycle) makes the control of the electric power converters a challenging topic in itself. In addition, the converter performances are tightly related to the adopted control method. Therefore, the control of electric power converters have attracted the attention of many researchers (Pinard, 2007; Rodriguez et al., 2000; Alvarez-Ramirez and Espinosa-Pérez, 2002).
Key Terms in this Chapter
Neimark-Sacker Bifurcation: The birth of a closed invariant curve from a fixed point in dynamical systems with discrete time (iterated maps), when the fixed point changes stability via a pair of complex eigenvalues with unit modulus. The bifurcation can be supercritical or subcritical, resulting in a stable or unstable (within an invariant two-dimensional manifold) closed invariant curve, respectively. When it happens in the Poincare map of a limit cycle, the bifurcation generates an invariant two-dimensional torus in the corresponding ODE.
Time Delayed Feedback Control: A controller introduced by K. Pyragas that intends to stabilize the periodic behavior of a chaotic system. The control action is proportional to the error between the present state and the T delayed state where T is the period of the intended behavior.
Fixed Point Induced Controller: A controller introduced by F. Angulo et al that intends to enlarge the stability region of a given behavior by reducing the the modulus of the corresponding Floquet multiplier. The idea is based on the use of the value of the fixed point if it is available.
H-Bridge Inverter: An electronic circuit that enables a voltage to be applied across a load in either direction. The main idea is to use four controlled electronic switches that toggle states pairwisely. Hence, the overall obtained dynamic is piecewise continuous and may exhibit a rich variety of behaviors regardless the type of load it is driving. These circuits are often used in robotics and other applications to allow DC motors to run forwards and backwards.
Flip Bifurcation: A period doubling bifurcation in a discrete dynamical system. It is a bifurcation in which the system switches to a new behavior with twice the period of the original system. That is, there exists two points such that applying the dynamics to each of the points yields the other point. Such bifurcation occurs when the corresponding Floquet multiplier quits the unit circle at z =-1of the complex plain when it varies with respect to a given system parameter denoted as the bifurcation parameter.
Quasi-Periodicity: An oscillating behavior that could appear when an oscillator is forced by an oscillating time-dependent input. Should the ratio of the intrinsic periodicity and the external period be irrational (incommensurate) then the behavior is called quasi periodic. However, if the ratio is rational then the behavior is rather periodic.
Border Collision Bifurcation: A non smooth flip bifurcation that occurs when the system dynamics hit a switching border.
Co-Existence of Behaviors: A special behavior that can happen in dynamic systems with multiple stable attractors. The limit behavior is determined by the initial state.