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SPECIAL REPORT
Written 1 October 2002; revised 25 February 2004
Chi Kong Tse
SummaryBifurcations and chaos have been studied in many power electronics circuits and systems. Numerous results have been reported regarding the nonlinear behaviour of such circuits and systems under variation of some selected parameters, e.g., period-doubling bifurcations, Hopf bifurcations, coexisting attractors, border collisions, etc. The current status of research in the identification of bifurcation behaviour in power electronics can be considered mature in the sense that the salient types of bifurcation behaviour, their underlying causes and the theoretical parameters affecting them have been well documented. Recently, research in this field has begun to seek possible applications that are of direct relevance to industrial power electronics. One direction is to apply some of the available research results in bifurcation behaviour to the design of practical power electronics systems. The main barrier is that the abstract mathematical presentations of the available results are not directly applicable to practical design problems. Our work in this research area has been directed to bridge that gap. The specific objectives of our work are
Towards the first objective, an almost complete re-examination of the problems associated with bifurcation phenomena in power electronics is needed. This is because much of the currently available results are expressed in terms of abstract mathematical conditions, which have no immediate connections with practical circuit conditions. In order for the results to be useful for engineering design, the research approach must necessarily incorporate a philosophical change in the way in which the problem is formulated. This leads naturally to the second objective stated above, which is to derive a practical design-oriented bifurcation analysis approach. Presently, most practical design rules are based on linearized models of the systems under study, and they take no consideration of possible (usually undesirable) bifurcations that can happen when some parameters are changed either accidentally or due to some design requirements. In contrast, the proposed design-oriented bifurcation analysis will lead to practical design rules that can facilitate realistic evaluations of performance in conjunction with the knowledge of bifurcation boundaries. Finally, it is still necessary to perform fundamental study of previously unknown bifurcation phenomena for some selected practical converter circuits. Moreover, the approach taken must be design-oriented so that the results can directly benefit the power electronics engineers.
I. IntroductionMuch of the work in the study of nonlinear phenomena of power electronics circuits and systems has been focused on basic research into the bifurcation and chaotic behaviour of power converters under variation of some selected parameters. Although the research results generated from these studies have greatly improved the understanding of the nonlinear behaviour of power electronics, their use in solving practical design problems is very limited and relatively unexplored, presumably due to the fundamental difference between the approaches taken for pursuing basic research and those required for facilitating practical design. Power electronics engineers frequently ask: what possible applications can the study of bifurcations and chaos offer to the practical design of industrial power electronics systems? Much of our research efforts has been (will be) devoted to showing the practical relevance of the study of bifurcations and chaos in power electronics. Specifically, we have attempted to pursue the following tasks:
The first task mentioned above is non-trivial and requires a complete reformulation of the problem. Specifically, currently available results are often given in terms of rather unrealistic circuit conditions and are based on idealised models which omit practical details. To make the results available for practical use, we need to include detailed circuit components and to consider practical operating scenarios. In most cases, the basic phenomena will remain the same, but the results become more significant if they are related to practical circuits (those actually in use) and are given in terms of realistic operating conditions. The second task is in effect a consolidation of existing analysis of bifurcations into a set of design-oriented procedures that can be directly applied to practical design problems. To gain further insights into the nonlinear behaviour of power electronics circuits and systems, efforts should be spent to uncover previously unknown phenomena and their underlying causes, in very much the same way as what basic researchers did in the past, but mainly for practical power electronics circuits. Therefore, the following task is complementary to the above two tasks.
II. Motivation and ImpactPower electronics as a discipline is driven by practical needs of industry. The continual quest for better products with ever increasing standards of reliability and performance makes it necessary to consider systematic approaches for performance optimization. Practical bifurcation analysis will have a long-term impact on the design of power electronics systems because the performance of a particular power electronics system is related to the choice of design parameters which are often bifurcation-sensitive. In reality, bifurcation is to be avoided, but it is also known that designing a system too remote from bifurcation boundaries may degrade performance characteristics such as transient speed. Therefore, design-oriented bifurcation analysis should have a significant impact on the practical methodologies taken to make design trade-offs and performance optimizations. III. Review of Current Research StatusPower electronics is a discipline spawned by real-life applications in industrial, commercial, residential and aerospace environments. Much of its development evolves around some immediate needs for solving specific power conversion problems. In the past three decades, this branch of electrical and electronic engineering has undergone an intense development in many areas of technology [1], including power devices, control methods, circuit design, computer-aided analysis, passive components, packaging techniques, and so on. The principal focus in power electronics, as reflected from topics discussed in some key conferences [2,3], is to fulfill the functional requirements of the application for which the circuits are used. Like many areas of engineering, power electronics is mainly motivated by practical applications, and it often turns out that a particular circuit topology or system implementation has found widespread applications long before it has been thoroughly analysed. For instance, the widespread application of a simple switching converter may date back to more than three decades ago. However, good analytical models allowing better understanding and systematic circuit design were only developed in the late 1970's [4], and in-depth analytical characterization and modelling is still being actively pursued today. Despite their common occurrence in power electronics circuits, nonlinear phenomena have only recently received appropriate formal treatments. Serious basic research in this area started mainly in the late 1980's [5]-[7], and much of the reported work has dealt with direct application of bifurcation theory to some appropriate nonlinear models of simple converter circuits [8]. Because simplified models and idealized operating conditions are considered, these basic works uncover the cause of many so-called "strange" nonlinear phenomena in power electronics, but fall short of any design-oriented analytical result that can directly benefit practicing engineers. In brief, the basic research of nonlinear phenomena in power electronics has enjoyed good academic recognition in the past, but is incompatible with the actual practice of power electronics. A. Review of Work Done by Other Researchers As mentioned above, serious work in this area began in the late 1980's and much of the work reported in the past has focused on fundamental study of nonlinear dynamical and chaotic behaviour of power converters. The following is a summary of work conducted by research groups other than ours at Hong Kong Polytechnic University. The occurrence of bifurcations and chaos in power electronics was first reported in the literature in the late 1980's by Hamill et al. [5]. Experimental observations regarding boundedness, chattering and chaos were also made by Krein and Bass [9] back in 1990. Although these early reports did not contain any rigorous analysis, they provided solid evidence of the importance of studying the complex behaviour of power electronics and its possible benefits for practical design. Since then, much interest has been directed towards pursuing formal studies of the complex phenomena commonly observed in power electronics. In 1990, Hamill et al. [5] reported a first attempt to study chaos in a simple buck converter at the IEEE Power Electronics Specialists Conference. Using an implicit iterative map, the occurrence of period-doublings, subharmonics and chaos in a simple buck converter was demonstrated by numerical analysis, PSPICE simulation and laboratory measurements. The derivation of a closed-form iterative map for the boost converter under a current-mode control scheme was presented later by the same group of researchers [10]. This closed-form iterative map allowed the analysis and classification of bifurcations and structural instabilities of this simple converter. Since then, a number of authors have contributed to the identification of bifurcation patterns and strange attractors in a wide class of circuits and devices of relevance to power electronics. Recently, an edited book which is devoted entirely to the subject of nonlinear phenomena in power electronics has also been published [11]. The bifurcation behaviour of the buck converter was studied by Chakrabarty et al. [12] who examined the bifurcation behaviour under variation of a range of circuit parameters including storage inductance, load resistance, output capacitance, etc. In 1996, Fossas and Olivar [13] presented a detailed analytical description of the buck converter dynamics, identifying the topology of its chaotic attractor and studying the regions associated with different system evolutions. Various possible types of operation of a simple voltage-feedback pulse-width-modulated buck converter were also investigated through the so-called stroboscopic map obtained by periodically sampling the system states. The bifurcation behaviour of dc/dc converters under current-mode control has been studied by a number of authors, e.g., Deane [14]. Typical bifurcation behaviour in power electronics contains transitions whereby a "sudden jump from periodic solutions to chaos" is observed. These transitions cannot be explained in terms of standard bifurcations such as "period-doubling" and "saddle-node". In fact, as proposed by Banerjee et al. [15,16] and Di Bernardo [17], these transitions are due to a class of bifurcations known as "border collisions", which is unique to switched dynamical systems [18]. Power electronics circuits other than dc/dc converters have also been examined in recent years. Dobson et al. [19] reported "switching time bifurcation" of diode and thyristor circuits. Such a bifurcation manifests as jumps in the switching times. Bifurcation phenomena from induction motor drives were reported separately by Kuroe [20] and Nagy et al. [21]. B. Review of Work Done at Hong Kong Polytechnic University The occurrence of period-doubling cascades for a simple dc/dc converter operating in discontinuous mode was first reported in 1994 by the author [22,23]. By modelling the dc/dc converter as a first-order iterative map, the onset of period-doubling bifurcations can be located analytically. The idea is based on evaluating the Jacobian of the iterative map about the fixed point corresponding to the solution undergoing the period-doubling, and determining the condition for which a period-doubling bifurcation occurs. Simulations and laboratory measurements have confirmed the findings. Formal theoretical studies of conditions for the occurrence of period-doubling cascades in a discontinuous-mode dc/dc converter were reported subsequently in [24]. For current-mode controlled converters, the author studied various types of routes to chaos and their dependence upon the choice of bifurcation parameters [25]. In 1995, the study of bifurcation phenomena was extended to a fourth-order Cuk dc/dc converter under a current-mode control scheme [26]. Moreover, when external clocks are absent and the system is "free-running" (for example, under a hysteretic control scheme), the system is autonomous and does not have a fixed switching period. A representative example is the free-running Cuk converter which has been shown [27] to exhibit Hopf bifurcation and chaos. Also, some attempts have been made to study higher order parallel-connected systems of converters which are becoming popular design choice for high-current applications [28,29]. Very recently, we began to look into applications of bifurcation analysis. Among the various specific applications, our work has been particularly fruitful in the study of a practical power-factor-correction converter and the application of bifurcation analysis to address a difficult fast-scale instability problem. The results have been given in terms of realistic circuit parameters and conditions, clearly defining the different stability regions of the parameter space [30,31]. IV. Methodologies for Future ResearchIn the previous section, we have identified the key problem in the development of applications of nonlinear dynamics and chaos theory to practical power electronics system design. This problem boils down to the general incompatibility between theoretical results in nonlinear analysis and the practical viewpoints of power electronics practitioners. To combat this problem, our basic methodology is to re-formulate the problems in bifurcations and chaos in terms of concepts that are already familiar to the engineer. To illustrate the idea, we take the classic current-mode controlled dc/dc converter as an example. The nonlinear analysis of this converter has uncovered interesting bifurcation patterns [14,25], but the results have not been useful to engineers. The difficulty in using the results or recognizing the usefulness of these results can be attributed to the way in which the analysis was performed. Traditionally, nonlinear analysis involves finding nonlinear models, dealing with differential equations (or iterative maps), computing Jacobians, and analysing stability of orbits [25]. The results are often stated in terms of abstract parameters, e.g., theoretical feedback gains and reference current levels. Although period-doubling bifurcations are well explained by such analysis, the power electronics community examined the same problem many years ago using a different viewpoint, examining mainly the gradient (slope) of the inductor current [32,33]. Effective methods for preventing period-doubling bifurcations known as "slope compensation" have already been used in practice for a long time. It transpires that if the analysis of this problem is re-formulated in terms of the familiar concept of slope compensation, results become more relevant to the design of these kinds of converters. Along this line, we have recently derived a practically relevant analysis for current-mode controlled converters [34]. This work has been further applied to power-factor-correction converters to explain a practical problem related to fast-scale instability [30,31]. The ease of analysis and the usefulness of the results depend very much on the choice of modelling methods. Thus, the choice of methodology for model building is crucial to this project. Our basic methodology is to employ a pre-analysis simulation and/or measurement. The purpose is to get an initial idea of the possible behaviour of the system under study. This is an important step because if the type of behaviour is approximately known, appropriate modelling methods can be chosen to fit the likely behaviour [8]. For example, it was once (incorrectly) believed that averaged modelling was useless for the study of chaos and bifurcations, and most studies were based on the more complicated discrete-time iterative mapping method. However, for systems that bifurcate from a regular high-frequency orbit to a long-period limit cycle (known as Hopf bifurcation), the expected phenomenon is essentially a low-frequency one and therefore can be captured by averaged models. If such a behaviour is known from a pre-analysis simulation or experiment, then averaged models would be the best choice for analysing the system as they are simple and adequate for capturing the salient low-frequency bifurcation behaviour [27]. Another important aspect of our future work that has a direct impact on the research methodology is the choice of the system for investigation. In the past, many basic research works are targeted at overly ideal systems for the sake of convenience or simplicity. The research methodology could then follow some previously developed theoretical procedures, and aim to achieve some closed-form analytical solution. For instance, in the study of border collisions, much has been done for the simple first-order case and rather detailed analytical results concerning the possible types of transitions have been reported [35,36]. In this project, we attempt to include as many practical components as possible in order to produce results that are directly relevant to practical design. In other words, we will refrain from the commonly used methodology of studying over simplified "ideal" cases. It is expected that significant new findings can be uncovered when practical (more complex) systems are considered. For example, we have recently identified intermittent chaos in power supplies that are constructed in practical "noisy" environments [37]. (Technically, the power supply under study takes into account unintended coupling of signals through the ground plane of the printed circuit board.) Finally, as mentioned before, we should target practical systems for investigation. It is therefore unlikely that the systems under study will lend themselves to simple low-order analysis. As a consequence, we will have to resort to numerical procedures and generality may be lost since particular parameter values must be assigned to perform numerical studies. Moreover, in order for the results to remain useful for the general case, they must be presented in terms of normalized parameters. V. Further ReadingThe work conducted by the Nonlinear Circuits and Systems Group at Hong Kong Polytechnic University in the area of complex behaviour of power converters has been expounded and reorganized in a recently published monograph [38], which also covers the basics of nonlinear system analysis and modelling of power electronics. References
Corresponding address: Professor Chi Kong Tse, Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hunghom, Hong Kong. Fax: +852 2362 8439. Email: cktse@ieee.org. Copyright © 2002, 2004 C. K. Tse. All rights reserved. |