John Alexander Taborda

EFFECTS OF QUANTIZATION, DELAY AND INTERNAL RESISTANCES IN DIGITALLY ZAD-CONTROLLED BUCK CONVERTER

Zero Average Dynamics (ZAD) strategy has been reported in the last decade as an alternative control technique for power converters, and a lot of work has been devoted to analyze it. From a theoretical point of view, this technique has the advantage that it guarantees fixed switching frequency, low output error and robustness, however, no high correspondence between numerical and experimental results has been obtained. These differences are basically due to model assumptions; in particular, all elements in the circuit were modeled as ideal elements and simulations and conclusions about steady state stability and transitions to chaos have been carried out with this ideal model. Regarding the practical point of view and the digital implementation, we include in this paper internal resistances, quantization effects and 1-period delay to the model. This paper shows in an experimental and numerical way the effects of these elements to the model and their incidence in the results. Now, experimental and numerical analyses fully agree.

BIFURCATION ANALYSIS ON NONSMOOTH TORUS DESTRUCTION SCENARIO OF DELAYED-PWM SWITCHED BUCK CONVERTER

In this paper, we propose a qualitative and quantitative dynamical study about the evolution of nonsmooth torus in a Digital Delayed Pulse-Width Modulator (PWM) switched buck converter. We explain the birth and destruction of the torus by successive discontinuity induced bifurcations (DIBs). The Digital-PWM control is based on Zero Average Dynamics (ZAD) strategy and a one-period delay is included in the control law. The control parameter (ks) of the ZAD strategy can be varied in a large range, ideally (-∞, ∞). On these borders, the dynamical behavior is the same and thus an annulus-like parameter space is considered. Under variation of ks, the system gets closer to a codimension-two bifurcation point, where two simultaneous border-collision bifurcations (BC) meet. The system leaves the high-order periodic behavior and a high-order band torus appears. Wrinkles and nonsmoothness due to more and more successive border-collisions bifurcations cause the torus destruction in high-order band chaos with tent-map-like structures and Mandelbrot-like sets. Finally, the chaotic bands merge in one-band chaos. The switched converter is modeled as a piecewise linear system where an analytical expression of the Poincare map is available. Characteristics as duality, symmetry and recurrent behavior are determined using additional numerical methods for the Poincare map decomposition in periodic sequences.

INTEGRATION-FREE ANALYSIS OF NONSMOOTH LOCAL DYNAMICS IN PLANAR FILIPPOV SYSTEMS

In this paper, we present a novel method to analyze the behavior of discontinuous piecewise-smooth autonomous systems (denominated Filippov systems) in the planar neighborhood of the discontinuity boundary (DB). The method uses the evaluation of the vector fields on DB to analyze the nonsmooth local dynamics of the Filippov system without the integration of the ODE sets. The method is useful in the detection of nonsmooth bifurcations in Filippov systems. We propose a classification of the points, events and events combinations on DB. This classification is more complete in comparison with the others previously reported. Additional characteristics as flow direction and sliding stability are included explicitly. The lines and the points are characterized with didactic symbols and the exclusive conditions for their existence are based on geometric criterions. Boolean-valued functions are used to formulate the conditions of existence. Different problems are analyzed with the proposed methodology.

Smooth bifurcations in 3D-parameter space of Digital-PWM Switched Converter

In this paper, we propose an analysis of smooth bifurcations in DC-DC buck converters controlled with digital-PWM. The three typical smooth bifurcations: Fold, Flip and Neimark-Sacker bifurcations are detected depending on control parameters. The presence of the three bifurcations in the same converter is not common and it has not been reported in Digital-PWM Switched Converters. Three parameters are varied in the Digital-PWM controller. Two parameters belong to a combined ZAD-FPIC (Zero Average Dynamic and Fixed Point Induced Control) control strategy and the third parameter is associated to Delays in the measured variables.

Computing and Controlling Basins of Attraction in Multistability Scenarios

The aim of this paper is to describe and prove a new method to compute and control the basins of attraction in multistability scenarios and guarantee monostability condition. In particular, the basins of attraction are computed only using a submap, and the coexistence of periodic solutions is controlled through fixed-point inducting control technique, which has been successfully used until now to stabilize unstable periodic orbits. In this paper, however, fixed-point inducting control is used to modify the domains of attraction when there is coexistence of attractors. In order to apply the technique, the periodic orbit whose basin of attraction will be controlled must be computed. Therefore, the fixed-point inducting control is used to stabilize one of the periodic orbits and enhance its basin of attraction. Then, using information provided by the unstable periodic orbits and basins of attractions, the minimum control effort to stabilize the target periodic orbit in all desired ranges is computed. The applicability of the proposed tools is illustrated through two different coupled logistic maps.

Adaptive Quasi-Sliding Mode Control for Permanent Magnet DC Motor

The motor speed of a buck power converter and DC motor coupled system is controlled by means of a quasi-sliding scheme. The fixed point inducting control technique and the zero average dynamics strategy are used in the controller design. To estimate the load and friction torques an online estimator, computed by the least mean squares method, is used. The control scheme is tested in a rapid control prototyping system which is based on digital signal processing for a dSPACE platform. The closed loop system exhibits adequate performance, and experimental and simulation results match.

Quantization Effects on Period Doubling Route to Chaos in a ZAD-Controlled Buck Converter

The quantization effect in transitions to chaos and periodic orbits is analyzed in this paper through a specific application, the zero-average-dynamics- (ZAD-) controlled buck power converter. Several papers have studied the quantization effects in the one periodic orbit and some authors have given guidelines to design digitally controlled power converter avoiding limit cycles. On the other hand many studies have been devoted to analyze the ZAD-controlled buck power converter, but these past studies did not include hardware considerations. In this paper, analog-to-digital conversion process is explicitly introduced in the modeling stage. As the feedback gain is varied, the dynamic behavior depending on the analog-to-digital converter resolution is numerically analyzed. Particularly, it is observed that including the quantizer in the model carries out several changes in the transitions to chaos, which include interruption of band-merging process by cascades of periodic inclusions, disappearing of band transitions, and multiple coexisting of periodic orbits. Many of these phenomena have not been reported as a consequence of the quantization effects.

CHARACTERIZATION OF CHAOS SCENARIOS WITH PERIODIC INCLUSIONS FOR ONE CLASS OF PIECEWISE-SMOOTH DYNAMICAL MAPS

In this paper, we study bifurcation scenarios characterized by period-adding cascades with alternating chaos in one class of piecewise-smooth maps (PWS). In this class, the state space is separated in three smooth zones defined by a saturation function. Some power converters controlled by Digital Pulse-Width Modulation (PWM) are physical applications of this class of PWS systems denoted by PWS3. Chaos has virtually been detected and studied in all disciplines, however the characterization problem of chaos scenarios has many open problems, mainly in nonsmooth dynamical systems. Novel bifurcation scenarios have recently been reported such as bandcount adding and bandcount increment scenarios based on the numerical detection of bands (where bands are considered as strongly connected components). However, this approach known as Bandcounter cannot be applied to detect bifurcations in chaos scenarios without crisis bifurcations or to identify topological changes inside of one-band chaos. We have proposed a novel framework named Dynamic Linkcounter approach to characterize chaos and torus breakdown scenarios in PWS systems. In this paper, we report overlapping period-adding cascades interspersed with a dynamic linkcount adding cascade. Each complex dynamic link (CDL) structure is a fingered strange attractor increasing in an arithmetic progression the number of CDL or fingers when a bifurcation parameter is varied. Alternative point of view based on tent-map-like structures is given to illustrate the formation of fingered strange attractors.

Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations

In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS) systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds—CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach.

Active Chaos Control of a Cam-Follower Impacting System using FPIC Technique

Abstract In this paper, we apply a novel active control scheme to suppress chaotic behavior in a cam-follower impacting system using Fixed-Point Inducting Control or FPIC Technique. Namely, the input speed of the cam-follower system is actively controlled in order to reach a single-impact periodic motion. A cam-follower system characterized by a radial cam and a flat-faced follower is used as a representative example. High-order periodic-impact and chaotic motions are controlled varying the input speed of the cam-follower system. Stroboscopic and impact maps are used to select the suitable control parameters under which stable single-impact periodic responses are reached.