Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems

This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. Several such codimension-one events have recently been identified, causing for example period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighbourhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the the grazing cycle is itself degenerate (e.g. non-hyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that have discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

Polynomial Curve Slope Compensation for Peak-Current-Mode-Controlled Power Converters

Linear ramp slope compensation (LRC) and quadratic slope compensation (QSC) are commonly implemented in peak-current-mode-controlled dc-dc converters in order to minimize subharmonic and chaotic oscillations. Both compensating schemes rely on the linearized state-space averaged model (LSSA) of the converter. The LSSA ignores the impact that switching actions have on the stability of converters. In order to include switching events, the nonlinear analysis method based on the Monodromy matrix was introduced to describe a complete-cycle stability. Analyses on analog-controlled dc-dc converters applying this method show that system stability is strongly dependent on the change of the derivative of the slope at the time of switching instant. However, in a mixed-signal-controlled system, the digitalization effect contributes differently to system stability. This paper shows a full complete-cycle stability analysis using this nonlinear analysis method, which is applied to a mixed-signal-controlled converter. Through this analysis, a generalized equation is derived that reveals for the first time the real boundary stability limits for LRC and QSC. Furthermore, this generalized equation allows the design of a new compensating scheme, which is able to increase system stability. The proposed scheme is called polynomial curve slope compensation (PCSC) and it is demonstrated that PCSC increases the stable margin by 30% compared to LRC and 20% to QSC. This outcome is proved experimentally by using an interleaved dc-dc converter that is built for this work

High-feedback Operation of Power Electronic Converters

electronic

Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems

Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth. Much of our understanding of these cases relies on a reduction to piecewise linearity near the border-collision. We also review a number of codimension-two bifurcations in which nonlinearity is important.Comment: pdfLaTeX, 9 figure

Dynamical analysis of an interleaved single inductor TITO switching regulator

We study the dynamical behavior of a single inductor two inputs two outputs (SITITO) power electronics DC-DC converter under a current mode control in a PWM interleaved scheme. This system is able to regulate two, generally one positive and one negative, voltages (outputs). The regulation of the outputs is carried out by the modulation of two time intervals within a switching cycle. The value of the regulated voltages is related to both duty cycles (inputs). The stability of the whole nonlinear system is therefore studied without any decoupling. Under certain operating conditions, the dynamical behavior of the system can be modeled by a piecewise linear (PWL)map, which is used to investigate the stability in the parameter space and to detect possible subharmonic oscillations and chaotic behavior. These results are confirmed by numerical one dimensional and two-dimensional bifurcation diagrams and some experimental measurements from a laboratory prototype.Peer ReviewedPostprint (published version

Bifurcation analysis of a DC–DC bidirectional power converter operating with constant power loads

Direct current (DC) microgrids (MGs) are an emergent option to satisfy new demands for power quality and integration of renewable resources in electrical distribution systems. This work addresses the large-signal stability analysis of a DC–DC bidirectional converter (DBC) connected to a storage device in an islanding MG. This converter is responsible for controlling the balance of power (load demand and generation) under constant power loads (CPLs). In order to control the DC bus voltage through a DBC, we propose a robust sliding mode control (SMC) based on a washout filter. Dynamical systems techniques are exploited to assess the quality of this switching control strategy. In this sense, a bifurcation analysis is performed to study the nonlinear stability of a reduced model of this system. The appearance of different bifurcations when load parameters and control gains are changed is studied in detail. In the specific case of Teixeira Singularity (TS) bifurcation, some experimental results are provided, confirming the mathematical predictions. Both a deeper insight in the dynamic behavior of the controlled system and valuable design criteria are obtained.Postprint (updated version

Bifurcation behavior of SPICE simulations of switching converters : a systematic analysis of erroneous results

Author name used in this publication: Chi K. Tse2007-2008 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Bifurcation analysis of a power-factor-correction boost converter : uncovering fast-scale instability

Author name used in this publication: Chi K. TseAuthor name used in this publication: Herbert H. C. luRefereed conference paper2002-2003 > Academic research: refereed > Refereed conference paperVersion of RecordPublishe

Virtual orbits and two-parameter bifurcation analysis in power electronic converters

When analytically investigating the 2-parameter bifurcation behaviour on a buck converter controlled by the ZAD strategy, the period doubling and corner collision curves in parameter space presented in [1] cross in four codimension two points presenting several properties. While two of them are given by the destruction of two bifurcation curves (period doubling and corner collision) at the boundaries of the feasible region in parameter space, a third one represents a change on the dynamics of the system and the fourth one is given by “appearance” of a saddle-node bifurcation curve. Using the concept of virtual and feasible orbits in the state space we will explain in this work how exactly this curve “appears” and we will add a fifth codimension two point where this saddle node bifurcation is destroyed and the corresponding curve “disappears”. We will also explain which are the changes on the dynamics of the system involved on the third codimension two point and we will give a partition of the parameter space so indicating where saturated and unsaturated T and 2T-periodic orbits exist. On the other hand, adding a third parameter representing a perturbation on the ZAD condition in one iteration of two periodic orbits, we will analytically show how some of the obtained phenomena are not possible to observe numerically due to the destruction of some of the previous structures for arbitrary small values of this parameter. At the same time, we will also justify why the numerical results when using an approximation of the ZAD condition (see [3]) contradict the analytical ones shown in [1]