Stabilization of oscillations through backstepping in high-dimensional systems

This paper introduces a method for obtaining stable and robust self-sustained oscillations in a class of single input nonlinear systems of dimension n ≥ 2. The oscillations are associated to a limit cycle that is produced in a second-order subsystem by means of an appropriate feedback law. Then, the controller is extended to the full system by a backstepping procedure. It is shown that the closed-loop system turns out to be generalized Hamiltonian and that the limit cycle can be thought as born in a Hopf bifurcation after moving a parameter.Ministerio de Ciencia e Innovación (España

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

Non-hyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach

Boundary equilibrium bifurcations in piecewise smooth discontinuous systems are characterized by the collision of an equi- librium point with the discontinuity surface. Generically, these bi- furcations are of codimension one, but there are scenarios where the phenomenon can be of higher codimension. Here, the possible col- lision of a non-hyperbolic equilibrium with the boundary in a two- parameter framework and the nonlinear phenomena associated with such collision are considered. By dealing with planar discontinuous (Filippov) systems, some of such phenomena are pointed out through specific representative cases. A methodology for obtaining the corresponding bi-parametric bifurcation sets is developed

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

Analysis of nonlinear and non-smooth dynamics of a self-oscillating series resonant inverter

In this paper, the dynamics of a dc-ac resonant self-oscillating LC series inverter is analyzed from the point of view of piecewise smooth dynamical systems. Our system is defined by two symmetric configurations and its bifurcation analysis can be given in a one dimensional param- eter space, thus finding a non smooth transition between two strongly different dynamics. The oscillating regime, which is the one useful for applications and involves a repetitive switching action between those configurations, is given whenever their open loop equilibrium is a fo- cus. Otherwise, the only attractors are equilibrium points of node type whose stable manifolds preclude the appearance of oscillations.Postprint (author's final draft

On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results.Ministerio de Ciencia e InnovaciónFondo Europeo de Desarrollo RegionalAgència de Gestió d'Ajuts Universitaris i de RecercaInstitució Catalana de Recerca i Estudis AvançatsJunta de Andalucí

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

The Focus-Center-Limit Cycle Bifurcation in Symmetric 3D Piecewise Linear Systems

The birth of limit cycles in 3D (three-dimensional) piecewise linear systems for the relevant case of symmetrical oscillators is considered. A technique already used by the authors in planar systems is extended to cope with 3D systems, where a greater complexity is involved. Under some given nondegeneracy conditions, the corresponding theorem characterizing the bifurcation is stated. In terms of the deviation from the critical value of the bifurcation parameter, expressions in the form of power series for the period, amplitude, and the characteristic multipliers of the bifurcating limit cycle are also obtained. The results are applied to accurately predict the birth of symmetrical periodic oscillations in a 3D electronic circuit genealogically related to the classical Van der Pol oscillator.Ministerio de Ciencia y Tecnología DPI2000-1218-C04-04Ministerio de Ciencia y Tecnología BFM2001-2668Ministerio de Ciencia y Tecnología BFM2003-00336Junta de Andalucía TIC-13