Two-Parameter Bifurcation Analysis of the Buck Converter

Colombo, Alessandro; Lamiani, Paola; Benadero, Luis; Bernardo, Mario di

doi:10.1137/080741434

Cited by 15 publications

(12 citation statements)

References 19 publications

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“…The results that we obtain, when compared with those included in [6], make evident the important role of DPWL systems as normal forms in the study of local bifurcations of piecewise smooth systems; see the final comments in section 4.…”

Section: Introductionmentioning

confidence: 87%

“…There seemingly appears a third codimension-two bifurcation point involving the two previous bifurcations in nondegenerate situations; see section 4. We conjecture that the two degenerate bifurcation points are linked by a curve which bounds the parameter region where two crossing limit cycles coexist, two being the maximum number of crossing periodic orbits.…”

Section: Introductionmentioning

confidence: 99%

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Canonical Discontinuous Planar Piecewise Linear Systems

Freire¹,

Ponce²,

Torres³

2012

SIAM J. Appl. Dyn. Syst.

188

142

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Abstract. The family of Filippov systems constituted by planar discontinuous piecewise linear systems with two half-plane linearity zones is considered. Under generic conditions that amount to the boundedness of the sliding set, some changes of variables and parameters are used to obtain a Liénard-like canonical form with seven parameters. This canonical form is topologically equivalent to the original system if one restricts one's attention to orbits with no points in the sliding set. Under the assumption of focus-focus dynamics, a reduced canonical form with only five parameters is obtained. For the case without equilibria in both open half-planes we describe the qualitatively different phase portraits that can occur in the parameter space and the bifurcations connecting them. In particular, we show the possible existence of two limit cycles surrounding the sliding set. Such limit cycles bifurcate at certain parameter curves, organized around different codimension-two Hopf bifurcation points. The proposed canonical form will be a useful tool in the systematic study of planar discontinuous piecewise linear systems, in which this paper is a first step.Key words. Filippov systems, normal form, limit cycle, piecewise linear differential systems AMS subject classifications. 34C05, 34C07, 37G15DOI. 10.1137/11083928X1. Introduction. Piecewise linear (PWL) systems constitute a class of differential systems which are widely used to model many real processes and different devices; see, for instance, [1,5,17]. The case of continuous PWL systems with two linearity regions separated by a straight line is the simplest possible configuration in piecewise smooth systems. Such a family of systems was completely studied in a previous paper [8], where in particular the existence of at most one limit cycle was established.Enforced by modern nonlinear engineering problems and mathematical biology (see [5]), some recent works in the literature on planar PWL systems deal with vector fields where continuity at the common boundary is not assumed. After the pioneering work of Filippov [7], one must cite in this context the papers of Coll, Gasull, and Prohens [3], Giannakopoulos and Pliete [9], Huan and Yang [11], Kuznetsov, Rinaldi, and Gragnani [13], Llibre, Ponce, and Torres [15], Shui, Zhang, and Li [18], and the recent thorough work of Guardia, Seara, and Teixeira [6], among others.As mentioned in the recent work of Huan and Yang [11], the study of discontinuous PWL systems is a difficult task because of the lack of a canonical form that can cope with a sufficiently broad class of systems, in contrast to what can be done for the continuous case; see [2]. In fact, only very particular cases are thoroughly analyzed in the available literature.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Canonical Discontinuous Planar Piecewise Linear Systems

Freire¹,

Ponce²,

Torres³

2012

SIAM J. Appl. Dyn. Syst.

188

142

View full text Add to dashboard Cite

show abstract

“…Examples include mechanical systems with friction and backlash, electrical and electronic circuits, walking and hopping robots and, more recently, biological and neural systems [19,20,2,3,4,14,21,8,7,9,26,5,10,15,1,6]. Many of these systems can be described by sets of piecewise smooth ordinary differential equations (ODEs) whose phase space is partitioned, by a set of switching manifolds, in different regions each associated to a different functional form of the system vector field.…”

Section: Introductionmentioning

confidence: 99%

Teixeira singularities in 3D switched feedback control systems

Colombo

Bernardo

Fossas

et al. 2010

Systems & Control Letters

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This paper is concerned with the analysis of a singularity that can occur in threedimensional discontinuous feedback control systems. The singularity is the two-fold -a tangency of orbits to both sides of a switching manifold. Particular attention is placed on the Teixeira singularity, which previous literature suggests as a mechanism for dynamical transitions in this class of systems. We show that such a singularity cannot occur in classical single-input single-output systems in the Lur'e form. It is, however, a potentially dangerous phenomenon in multiple-input multiple-output switched control systems. The theoretical derivation is illustrated by means of a representative example.

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“…The second model (5) is the exact discrete time model which has the advantage of carrying out fast numerical simulations but it presents the same shortcoming of the switched model in terms of mathematical treatment. The third model (14) which comes from tenable simplification of the exact discrete time model presents several advantages. First of all, the model is unique for all operating continuous conduction mode.…”

Section: Model Verification From Numerical Simulationsmentioning

confidence: 99%

“…Analysis, characterization and modeling of simple switching converters are still actively pursued today [13,14]. It has been discovered that despite their simplicity, power converters are copious in nonlinear dynamics.…”

Section: Introductionmentioning

confidence: 99%