Gerard Olivar

Study of chaos in the buck converter

A DC-DC buck converter controlled by naturally-sampled, constant-frequency pulsewidth modulation in continuous conduction mode gives rise to a great variety of behavior, depending on the values of the parameters of the circuit. We analyze the one-periodic and two-periodic orbits which cross the voltage ramp once per cycle, and we study their stability by computing the characteristic multipliers associated with each one. Subharmonics, bifurcations, and the presence of a strange attractor are also studied. A plot of the number of crossings in the ramp is drawn. This becomes a helpful tool for investigating the evolution of the trajectories when they are close to the attractor. When analytic computations are impossible, we resort to numerical algorithms to simulate the orbits.

Stabilization of periodic orbits of the buck converter by time‐delayed feedback

Time-delay autosynchronization (TDAS) can be used to stabilize unstable periodic orbits in dynamic systems. The technique involves continuous feedback of signals delayed by the orbits period so that the feedback signal vanishes on the target orbit and hence the latter is a solution of the original dynamic system. Furthermore, this control method only requires the knowledge of the period of the unstable orbit. The feedback gain needed to achieve stabilization varies with the bifurcation parameter(s) of the system, resulting in a domain of control, the computation of which requires, in general, detailed information about the target orbit(s). In this paper we compute the domain of control of the unstable periodic orbits of the PWM controlled buck converter for a couple of TDAS schemes. For both schemes we get an analytical expression for the closed curve whose index determines the stability, and this index is then numerically computed. We run several simulations of the controlled systems and discuss the results. The main result is that TDAS greatly increases the range of values of the input voltage where the PWM control yields a periodic orbit with a small rippling. Copyright

SECONDARY BIFURCATIONS AND HIGH PERIODIC ORBITS IN VOLTAGE CONTROLLED BUCK CONVERTER

Period doubling route to chaos has been shown to occur in the voltage controlled DC/DC buck converter, both experimentally and numerically. A chaotic attractor was found at the end of the sequence, suddenly followed by an increase of its size. In this paper new secondary bifurcations and high periodic phenomena, coexisting with the main sequence are detected and analyzed over the same range of parameters. A(synchronous)-switching and stroboscopic maps, unstable orbits, bifurcation diagrams, invariant manifolds and basins of attraction are outlined. These tools are put together to reveal the dynamical richness of this nonsmooth system.

TRANSITION FROM PERIODICITY TO CHAOS IN A PWM-CONTROLLED BUCK CONVERTER WITH ZAD STRATEGY

The transition from periodicity to chaos in a DC-DC Buck power converter is studied in this paper. The converter is controlled through a direct Pulse Width Modulation (PWM) in order to regulate the error dynamics at zero. Results show robustness with low output error and a fixed switching frequency. Furthermore, some rich dynamics appear as the constant associated with the first-order error dynamics decreases. Finally, a transition from periodicity to chaos is observed. This paper describes this transition and the bifurcations in the converter. Chaos appears in the system with a stretching and folding mechanism. It can be observed in the one-dimensional Poincare map of the inductor current. This Poincare map converges to a tent map with the variation of the system parameter ks.

Feedback control of limit cycles: a switching control strategy based on nonsmooth bifurcation theory

In this paper, we present a method to control limit cycles in smooth planar systems making use of the theory of nonsmooth bifurcations. By designing an appropriate switching controller, the occurrence of a corner-collision bifurcation is induced on the system and the amplitude and stability properties of the target limit cycle are controlled. The technique is illustrated through a representative example.

Two-Parameter Discontinuity-Induced Bifurcation Curves in a ZAD-Strategy-Controlled DC–DC Buck Converter

The dynamics of a zero-average dynamic strategy controlled dc-dc Buck converter, modelled by a set of differential equations with discontinuous right-hand side is studied. Period-doubling and corner-collision bifurcations are found to occur close to each other under small parameter variations. Closer examination of the parameter space leads to the discovery of a novel bifurcation. This type of bifurcation has not been reported so far in the literature and it corresponds to a corner-collision bifurcation of a nonhyperbolic cycle. The bifurcation boundaries are computed analytically in this paper and the system dynamics are unfolded close to the novel bifurcation point.

STABILIZING A TWO-CELL DC-DC BUCK CONVERTER BY FIXED POINT INDUCED CONTROL

In this paper, we study nonlinear and bifurcation behavior of a two-cell DC-DC buck power electronic converter. The system shows nonsmooth period doubling bifurcation and chaotic phenomena in a certain zone of parameter space. This zone is located both analytically and from numerical simulations. One-dimensional, two-dimensional bifurcation diagrams and Lyapunov exponent spectrum are used to detect the different dynamic behaviors of the system. The Fixed Point Induced Control (FPIC) technique is applied to the system in order to widen the stability zone. The performance of the FPIC technique applied to the stabilization of a two-cell DC-DC buck converter is analyzed. With this technique, stabilization is achieved without changing the fixed point. The robustness in the presence of a noisy environment is checked by numerical simulations by considering different noise levels.

Bounding the Output Error in a Buck Power Converter Using Perturbation Theory

We show the main results obtained when applying the average theory to Zero Average Dynamic control technique in a buck power converter with pulse-width modulation (PWM). In particular, we have obtained the bound values for output error and sliding surface. The PWM with centered and lateral pulse configurations were analyzed. The analytical results have confirmed the numerical and experimental results already obtained in previous publications. Moreover, through an important lemma, we have generalized the theory for any stable second-order system with relative degree 2, using properties related to transformations and stability of linear systems.

Chaos stabilization with TDAS and FPIC in a buck converter controlled by lateral PWM and ZAD

In this paper the performance of TDAS (time-delay autosynchronization) and FPIC (fixed point induced control) techniques, controlling chaos in the buck converter is analyzed. With these techniques, the stabilization of the 1-periodic orbit in the buck converter is controlled by lateral PWM (pulse width modulation) and ZAD (zero average dynamics) strategies.

Time-delay stabilization of the buck converter

Time-delay autosynchronization (TDAS) can be used to stabilize unstable periodic orbits in dynamical systems. The technique involves continuous feedback of signals delayed by the orbits period so that the feedback signal vanishes on the target periodic orbit and hence the stabilized periodic orbit is a solution of the original dynamical system. Furthermore, this control method only requires the knowledge of the period of the unstable orbit. The amount of feedback gain needed to achieve stabilization varies with the bifurcation parameter(s) of the system, resulting in a domain of control. We compute the domain of control of the unstable periodic orbits of the buck converter for a given TDAS scheme. We obtain a closed analytical expression for the curve g:S/sup 1//spl rarr/C whose index determines the stability, and this index is then numerically computed. We run several simulations of the controlled system and discuss the results.