Canonical Discontinuous Planar Piecewise Linear Systems

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 Li´enard-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

On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

In this paper we study the non-existence and the uniqueness of limit cycles for the Liénard differential equation of the form x'' − f(x)x' + g(x) = 0 where the functions f and g satisfy xf(x) > 0 and xg(x) > 0 for x ≠ 0 but can be discontinuous at x = 0. In particular, our results allow us to prove the non-existence of limit cycles under suitable assumptions, and also prove the existence and uniqueness of a limit cycle in a class of discontinuous Liénard systems which are relevant in engineering applications

Existencia y unicidad de soluciones periódicas para una ecuación de Liénard discontinua lineal a trozos

En esta comunicación se establece un resultado de existencia y unicidad de soluciones periódicas en una ecuación de tipo Liénard, donde las funciones involucradas son lineales a trozos discontinuas. Para ello, se ha transformado la ecuación inicial en un sistema plano de Liénard y se ha seguido el método convexo de Filippov para extender las órbitas que alcanzan la línea de discontinuidad. Nos hemos limitado a considerar sistemas que no poseen soluciones deslizantes (sliding motions) en el sentido de Filippov

Delay effects on the limit cycling behavior in an H-bridge resonant inverter with zero current switching control strategy

Celebrado en Tarragona del 2-6 de septiembre de 2018.In this paper, bifurcations of limit cycles in a H-bridge LC resonant inverter under a zero current switching control strategy with delay in the switching action are analyzed. Mathematical analysis and numerical simulations show that the delay can degrade the quality of the oscillations and even inhibit them.Agencia Estatal de Investigación DPI2017- 84572-C2-1-RFondo Europeo de Desarrollo Regional DPI2017- 84572-C2-1-RMinisterio de Ciencia e Innovación MTM2015-65608-PJunta de Andalucía Consejería de Economía y Conocimiento P12-FQM-165

Limit cycle bifurcations in resonant LC power inverters under zero current switching strategy

The dynamics of a DC-AC self-oscillating LC resonant inverter with a zero current switching strategy is considered in this paper. A model that includes both the series and the parallel topologies and accounts for parasitic resistances in the energy storage components is used. It is found that only two reduced parameters are needed to unfold the bifurcation set of this extended system: one is related to the quality factor of the LC resonant tank, and the other accounts for the balance between serial and parallel losses. Through a rigorous mathematical study, a complete description of the bifurcation set is obtained and the parameter regions where the inverter can work properly is emphasized.Postprint (author's final draft

Dynamic analysis of self-oscillating H-bridge inverters with state feedback

This paper presents a comprehensive approach to analyze the dynamics of a generalized model of resonant inverters using nonsmooth dynamical system theory. The model simultaneously covers both parallel and series resonant inverters under state feedback control. The multi-parametric physical space is reduced to a plane, which is divided in several regions with different dynamical behavior. The boundaries separating these regions are located by solving their corresponding equations and it is found that they all emerge from a singular point in the parameter plane. Suitability for applications of these regions is emphasized, thus providing useful criteria for parameter selection.Postprint (author's final draft

Bifurcation patterns in homogeneous area-preserving piecewise-linear maps

The dynamical behavior of a family of planar continuous piecewise linear maps with two zones is analyzed. Assuming homogeneity and preservation of areas we obtain a canonical form with only two parameters: the traces of the two matrices defining the map. It is shown the existence of sausage-like structures made by lobes linked at the nodes of a nonuniform grid in the parameter plane. In each one of these structures, called resonance regions, the rotation number of the associated circle map is a given rational number. The boundary of the lobes and a significant inner partition line are studied with the help of some Fibonacci polynomials.Postprint (author's final draft

Suppression of undesired attractors in a self-oscillating H-bridge parallel resonant converters under zero current switching control

Resonant converters under zero current switching control strategy can exhibit coexistence of attractors, making it difficult the startup of the system from zero initial conditions. In this paper, the problem of multiple coexisting attractors in parallel resonant converters is addressed. Appropriate modifications of the switching decision with the aim of converting undesired attractors into virtual ones are proposed. A suitable control signal is generated from the state variables of the system and used to adjust the switching decision. Numerical simulations corroborate the proposed solutions and the simplest one was finally verified by measurements from a laboratory prototype.Postprint (author's final draft

Nonlinear dynamic modeling and analysis of self-oscillating H-bridge parallel resonant converter under zero current switching control: unveiling coexistence of attractors

This paper deals with the global dynamical analysis of an H-bridge parallel resonant converter under a zero current switching control. Due to the discontinuity of the vector field in this system, sliding dynamics may take place. Here, the sliding set is found to be an escaping region. Different tools are combined for studying the stability of oscillations of the system. The desired crossing limit cycles are computed by solving their initial value problem and their stability analysis is performed using Floquet theory. The resulting monodromy matrix reveals that these cycles are created according to a smooth cyclic-fold bifurcation. Under parameter variation, an unstable symmetric crossing limit cycle undergoes a crossing-sliding bifurcation leading to the creation of a symmetric unstable sliding limit cycle. Finally, this limit cycle undergoes a double homoclinic connection giving rise to two different unstable asymmetric sliding limit cycles. The analysis is performed using a piecewise-smooth dynamical model of a Filippov type. Sliding limit cycles divide the state plane in three basins of attraction, and hence, different steady-state solutions may coexist which may lead the system to start-up problems. Numerical simulations corroborate the theoretical predictions, which have been experimentally validated.Postprint (author's final draft

Delay effects on the limit cycling behavior in an H-bridge resonant inverter with zero current switching control strategy

In this paper, bifurcations of limit cycles in a H-bridge LC resonant inverter under a zero current switching control strategy with delay in the switching action are analyzed. Mathematical analysis and numerical simulations show that the delay can degrade the quality of the oscillations and even inhibit them.Postprint (author's final draft