Dynamic Analysis And Qft-based Robust Control Design Of Switched-mode .
Helsinki University of Technology Control EngineeringEspoo 2008Report 157DYNAMIC ANALYSIS AND QFT-BASED ROBUST CONTROLDESIGN OF SWITCHED-MODE POWER CONVERTERSAli Al.TowatiTEKNILLINEN KORKEAKOULUTEKNISKA HÖGSKOLANHELSINKI UNIVERSITY OF TECHNOLOGYTECHNISCHE UNIVERSITÄT HELSINKIUNIVERSITE DE TECHNOLOGIE D HELSINKI
Helsinki University of Technology Control EngineeringEspoo September 2008Report 157DYNAMIC ANALYSIS AND QFT-BASED ROBUST CONTROLDESIGN OF SWITCHED-MODE POWER CONVERTERSAli Al.TowatiDissertation for the degree of Doctor of Science in Technology to be presented with due permission ofthe Faculty of Electronics, Communications and Automation, for public examination and debate inAuditorium AS1 at Helsinki University of Technology (Espoo, Finland) on the 21st of October, 2008,at 12 noon.Helsinki University of TechnologyFaculty of Electronics, Communications and AutomationDepartment of Automation and Systems Technology
Distribution:Helsinki University of TechnologyDepartment of Automation and Systems TechnologyP.O. Box 5500FI-02015 TKK, FinlandTel. 358-9-451 5201Fax. 358-9-451 5208E-mail: N 978-951-22-9574-6 (printed)ISBN 978-951-22-9575-3 (pdf)ISSN 0356-0872YliopistopainoHelsinki 2008Available on net at http://lib.tkk.fi/Diss/2008/isbn9789512295753
ABHELSINKI UNIVERSITY OF TECHNOLOGYP.O. BOX 1000, FI-02015 TKKhttp://www.tkk.fiABSTRACT OF DOCTORAL DISSERTATIONAuthorAli Al.TowatiName of the dissertationDynamic Analysis and QFT-Based Robust Control Design of Switched-Mode Power ConvertersManuscript submitted11.6.2008Date of the defence21.10.2008Manuscript revised 8.9.2008MonographArticle dissertation (summary original articles)FacultyElectronics, Communications and AutomationDepartmentAutomation and Systems TechnologyField of researchControl EngineeringOpponent(s)Prof. Riku Pöllänen and Dr. Mikko HankaniemiSupervisorProf. Heikki KoivoInstructorsProf. Teuvo Suntio and Dr. Kai ZengerAbstractThe use of switched-mode power converters is continuously growing both in power electronics products and systems, e.g.in Telecom applications, commercial grid systems etc.The switching converters are required to provide robust behavior and to operate without instability under a variety ofoperation conditions. Hence the converter system may be subject to disturbances due to load, input voltage, and systemparameter variations. In the thesis a robust control design procedure based on the QFT method (Quantitative FeedbackTheory) is applied successfully for switching-mode DC-DC converters in order to achieve robust output in spite ofdifferent uncertainties. Simulation results are presented to demonstrate and validate the control design, showing gooddynamic performance of the QFT controller.When designing large-scale systems it is often impractical to analyze and design the system as a whole. Instead, it isdesirable to divide the system into manageable subsystems which can then be designed independently. The subsystemsmay then be connected together to form a complete integrated system. One of the major difficulties in integratedsubsystems is the stability performance degradation due to the interaction between the subsystems.A formalism to analyze the interaction between subsystems using the unterminated two-port small-signal representation isderived. Two-port models are first defined as unterminated models, where the effect of load is excluded but may be easilyincluded using the developed reflection rules. The use of the impedance ratio as a minor loop gain, which can be used tocheck system stability, is outlined.Recently, there has been increasing interest in the parallel operation of DC-DC converters for reasons of increasing systemreliability, facilitating system maintenance, allowing for future expansion, and reducing system design cost. However,paralleled DC-DC converters require a systematic modeling methodology and a categorical current-sharing mechanism toimprove a performance of the overall system.In order to achieve desirable characteristics when operating converter modules in parallel, a unified systematic approachedfor modeling of parallel DC-DC converter with current-sharing control, is proposed, developed, and analyzed.KeywordsSwitched-mode converters, QFT-based robust control, subsystem interaction, current-sharing control.ISBN (printed)978-951-22-9574-6ISSN (printed)ISBN (pdf)978-951-22-9575-3ISSN (pdf)LanguageEnglishNumber of pages 145PublisherHelsinki University of Technology, Department of Automation and Systems TechnologyPrint distributionHelsinki University of Technology, Department of Automation and Systems TechnologyThe dissertation can be read at 0872
PrefaceThis research work has been carried out at the Department of Automation and SystemsTechnology of Helsinki University of Technology.First of all I would like to thank my God for blessing me with the ability to completethis work successfully.I am indebted to my supervisor Professor Heikki Koivo for giving me the opportunity towork in his laboratory and providing me with an excellent atmosphere for doing research.I am deeply grateful to my instructor, Professor Teuvo Suntio from Tampere University ofTechnology for his invaluable guidance and patience during the process. I wish to expressmy sincere appreciation to D.Sc. Kai Zenger for his support and all the time he spent withme discussing this work. I am grateful to the pre-examiners Prof. Pertti Silventoinen andDr. Mikko Hankaniemi for their valuable comments and recommendations.I would also like to thank all the people in the Control Engineering Group for creatingan enjoyable atmosphere to work. The Academy of Finland and the Research Foundation ofHelsinki University of Technology through different technology programs have supported thisresearch financially, which are gratefully acknowledged. In addition, the grants received fromFinnish Society of Automation, Elektroniikkainsinöörien Säätiö, Finnish Cultural Foundation and Alfred Kordelin Säätiö are gratefully acknowledged.Deepest gratitude to my family and relatives for their continuous support. Finally, mydearest thanks go to my wife Salha, my daughters Fatma and Arwa for providing me withtheir everlasting love and confidence. Great thanks to all friends here in Finland and inLibya or elsewhere for their care and commitment.Espoo, September, 2008Ali Al.Towativ
“This work is dedicated to the memory of my father, Mohammed, who has always been verysupportive, patient, understanding, and encouraging. To the memories of my dearest brother,Elmuntaser and my beloved sister, Nafeesa, who have both passed away a few months ago,you will always have a place in my heart. It is also dedicated to my mother Fatma, for hercontinuous love, support, and encouragement”vii
List of GciGcidGcLGcoGcscGDGio oGic oGiL oGio cGio fGjc oGjL oGaHseH HvState-space realization of a linear systemQFT boundControl variableCapacitor or capacitanceFilter capacitanceDuty cycleSteady state duty cycleDiodeOutput disturbance signalVoltage-source in current-output converterPrefilterFilter resonance frequencyCrossover frequencyThe resonant frequency of output averaging filterFeedforward gain from the input voltageSwitching frequencyDuty cycle gainFeedback gain from the output voltageController transfer functionControl-to-input transfer functionCross-coupling transfer functionTransfer function from control-to-inductor currentControl-to-output transfer functionCurrent-sharing controller transfer functionOutput disturbance modelOpen-loop line-to-output transfer functionTransfer function from line-to-capacitor voltageTransfer function from line-to-inductor currentClosed-loop line-to-output transfer functionFilter forward-voltage transfer functionTransfer function from output current-to-capacitor voltageTransfer function from output-to-inductor currentGain factor matching the voltage control signal to the internalcontrol signalSensor gainHardy space of transfer functions with bounded -normVoltage-sensing gainix
List of SymbolsiCicoiiniinciLipioIinILIoj(s)jNjoJoKLLg (s)Lgnom (s)Lc (s)Lco (s)Lcsc (s)Lm (s)Lv (s)m1m2M tCapacitor currentControl commandInput currentBus currentInductor currentPeak of inductor currentOutput currentDC-value of input currentDC-value of inductor currentDC-value of output currentGain of controlled current source in canonical equivalent circuitNorton current sinkLoad current sinkDC-value of load current sinkController gainInductor or inductanceLoop gainNominal loop gainCurrent loop gainCurrent-output loop gainCurrent-sharing loop gainMinor loop gainVoltage loop gainInductor current slope when the switch is ONInductor current slope when the switch is OFFVoltage conversion ratioCompensating ramp slopeMaximum peak magnitudemeasured noiseNumber of dc-dc converters in parallelCharge received by capacitorQuality FactorPlant modelConverter input powerConverter output powerEquivalent series resistance of capacitor of the converterEquivalent series resistance of capacitor of the filterThe dynamic resistance associated to diodeThe MOSFET on-time channel resistanceEquivalent series resistance of inductor of the converterEquivalent series resistance of inductor of the filterResistance or resistorEquivalent resistanceConverter input resistanceResistive loadCurrent-sensing resistorSwitchSensitivity functionTime in secondsx
List of SymbolstontofTTji oTji cTji fTLTsTUucuCuCFuduLuinuincuourUUCUDUinUoVstWs (s)XyYYin Yin oYin cYin fYin scYo cscZinZin oZin cZLZLvZoZo oZo cZo fZsΔiLγωBON-time of the switchOF-time of the switchComplementary sensitivity functionOpen-loop input susceptibility to load changesClosed-loop input susceptibility to load changesFilter reverse-current transfer functionLower tracking boundSwitching time intervalUpper tracking boundOutput of voltage controllerOutput capacitor voltageFilter output capacitor voltagePlant inputInductor voltageInput voltageFilter output voltageOutput voltageReference signalInput vectorDC-value of capacitor voltageVoltage loss of the diodeDC-value of input voltageDC-value of output voltageSawtooth waveform amplitude, i.e. PWM gainWeighting functionState-variable vectorPlant outputOutput vectorInput admittance of ideally controlled converterOpen-loop input admittanceClosed-loop input admittanceFilter input admittanceInput admittance of nulled output voltageCurrent-sharing output admittance of slave moduleInput impedanceOpen-loop input impedanceClosed-loop input impedanceLoad impedanceImpedance of parallel-connected output capacitor and loadimpedanceOutput impedanceOpen-loop output impedanceClosed-loop output impedanceFilter output impedanceSource impedanceThe difference between a peak inductor current and its averagedRobust stability boundBandwidth frequencyxi
List of AbbreviationsACAC/DCBWCCMCSDCDC ing currentAC to DC rectifierBandwidthContinuous conduction modeCurrent sharingDirect currentDC to DC converterDiscontinuous-conduction modeDistributed power architectureDistributed power supplyDigital signal processorExtra-element-theoremElectromagnetic interferenceEquivalent series resistanceGain marginGain margin and phase margin criterionGraphical user interfaceKirchhoff’s Current LawKirchhoff’s Voltage sistanceLinear fractional transformationLinear time invariantLeft-hand planeLinear Quadratic GaussianMulti input/multi outputMetal-oxide-semiconductor field-effect transistorMaster-Slave ControlNegative-resistance-oscillatorPeak current-mode controlProportional-integral controlPower factor correctionPhase marginPulse-width modulatorQuantitative feedback theoryRight-hand plane zeroxiii
List of AbbreviationsSISOSMPSSSATODFVMCSingle input/single outputSwitching-mode power supplyState-space averagingTwo degree of freedomVoltage-mode controlxiv
ContentsAbstractiiiPrefacevList of SymbolsixList of AbbreviationsxiiiContentsxv1 Introduction1.1 Background . . . . . . . . . . . . . . . . . . . . .1.2 Modeling of Switching DC-DC Converters . . . .1.3 Control Design of Switching DC-DC Converters .1.4 Stability and Subsystems Interactions . . . . . .1.5 Modeling and Dynamics Analysis of Multimodule1.6 Research Objectives . . . . . . . . . . . . . . . .1.7 Outline of the Thesis . . . . . . . . . . . . . . . .1.8 Thesis’s Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .DC-DC Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1112333442 DC-DC Switching Converters2.1 The Buck DC-DC Converter . . . . . . . . . . .2.2 The Boost DC-DC Converter . . . . . . . . . . .2.3 The Buck-Boost DC-DC Converter . . . . . . . .2.4 Modes of Operation of the DC-DC Converter . .2.4.1 Continuous Conduction Mode CCM . . .2.4.2 Discontinuous Conduction Mode DCM . .2.5 Control Structures of DC-DC Converter . . . . .2.5.1 Voltage-Mode Control VMC . . . . . . . .2.5.2 Peak Current-Mode Control PCMC . . .2.6 Modeling of DC-DC Switching Power Converters2.6.1 Continuous Conduction Mode CCM . . .2.6.2 Discontinuous Conduction Mode DCM . .2.7 Modeling of Pulse-Width Modulator . . . . . . .2.7.1 Voltage-Mode PWM . . . . . . . . . . . .2.7.2 Peak Current-Mode PWM . . . . . . . . .788991010101111111215202025xv.
CONTENTS3 Robust Control Design for Switching-Mode Power Converters3.1 Quantitative Feedback Theory (QFT) . . . . . . . . . . . . . . . . . . . . . .3.1.1 Closed-Loop Formulation . . . . . . . . . . . . . . . . . . . . . . . . .3.1.2 Robust Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.3 Uncertainty Model and Plant Templates . . . . . . . . . . . . . . . . .3.1.4 QFT Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.5 QFT Design for Uncertain Non-minimum Phase Systems . . . . . . .3.2 QFT-Based Robust Controller Design for Switching-Mode Power Converters .3.2.1 Voltage-Mode-Controlled Converter . . . . . . . . . . . . . . . . . . .3.2.2 Peak-Current-Mode-Controlled Converter . . . . . . . . . . . . . . . .3.3 QFT-Based Robust Controller Design for Non-minimum Phase Converters . .3.3.1 QFT-Based Robust Controller Design for a Boost Converter . . . . .3.3.2 QFT-Based Robust Controller Design for a Buck-Boost Converter . .333434363839414343576666784 Subsystem Interaction Analysis4.1 Two-Port Network . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.1 Unterminated Modeling Approach . . . . . . . . . . . . . .4.2 System Stability and Performance . . . . . . . . . . . . . . . . . .4.2.1 Linear fractional transformations: The matrix star product4.2.2 Internal Stability . . . . . . . . . . . . . . . . . . . . . . . .4.2.3 Forbidden region concept . . . . . . . . . . . . . . . . . . .4.3 Load and Supply Interaction Analysis . . . . . . . . . . . . . . . .4.3.1 Load Interaction Analysis . . . . . . . . . . . . . . . . . . .4.3.2 Source Interaction Analysis . . . . . . . . . . . . . . . . . .4.4 Input Filter Interactions in Switched-Mode Power Converters . . .4.4.1 EMI Filters for Switching-Mode Power Converters . . . . .4.4.2 Nature of the Oscillation Problem . . . . . . . . . . . . . .4.4.3 Application of Two-Port Representation . . . . . . . . . . .8586868787899091921011021021041055 Dynamics Analysis of Paralleled DC-DC Converters5.1 General Constraints on Paralleling DC-DC Converters . . . . . . .5.1.1 Current-Output Converters . . . . . . . . . . . . . . . . . .5.1.2 Equivalent Circuit Models for DC-DC Switching Converters5.2 Paralleled DC-DC Converters with Master-Slave Control MSC . .5.2.1 Modeling of Multimodule Converters with MSC . . . . . . .5.2.2 QFT-Based Robust Controller Design . . . . . . . . . . . .1171181191201211211256 Conclusions1356.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137BibliographyA139AppendixA.1 MatlabTM /Simulink Simulation Setup . . . . . . . . . . . .A.1.1 Single Output-Voltage DC-DC Converters . . . . .A.1.2 Single Output-Current DC-DC Converters . . . . .A.1.3 Multimodule DC-DC Parallel Converters with MSCxvi.A-1A-1A-1A-6A-7
Chapter1Introduction1.1 BackgroundThe electric power is not normally used in the form in which it was produced or distributed.Practically all electronic systems require some form of power conversion. A device whichtransfers electric energy from the source to the load using electronic circuits is called aPower Supply, although power converter would be a more accurate term for such a device.A typical application of a power supply is to convert utility AC voltage into regulated DCvoltages required for electronic equipment. Nowadays in most power supplies providingmore than a few watts the energy flow is controlled with power semiconductors that arecontinuously switching on and off with high frequency. Such devices are called Switch ModePower Supplies or SMPS. In general, SMPS can be classified into four types according tothe form of input and output voltages: AC to DC (off-line power supply or a rectifier);DC to DC (voltage converter); AC to AC (frequency changer or cycloconverter); DC to AC(inverter). In this thesis, the modeling, control design challenges and subsystems interactionissues will be treated only for DC-DC converters.Switching-mode power-electronic converters are nonlinear dynamical systems. The nonlinearities arise primarily due to switching, power devices, and passive components such asinductors, capacitors and parasitics. SMPS’s represent different circuit topologies or configurations within each switching cycle. For the continuous conduction mode, there are twotopologies. For the discontinuous conduction mode of operation, a third configuration hasto be added to yield a total of three topologies. In each configuration, the system can bedescribed by linear state equations. Switching between the different topologies will varyfrom cycle to cycle depending on the output of the system, and this complicates the analysisfurther.The static conservation properties of the elementary switching converters (buck, boost,and buck-boost) have been thoroughly understood since the early 1970s. This is one of themain reasons of their ever-increasing number of applications in electrical energy conversion.However, the complete dynamics behavior of switching power converters still has to befurther understood and improved. This is not possible without an in-depth understandingof the operation of such circuits and without easy-to-use and accurate models.1.2 Modeling of Switching DC-DC ConvertersModeling and analysis of switching DC-DC converters can be either numerical or analytical.In numerical techniques, several algorithms or circuit simulators are used to produce quan1
Chapter 1Introductiontitative results. These methods are easy to use. They posses accuracy and universality andthey are applicable when no equivalent model is available. However, they are fail to providethe design insight needed to understand the behavior of switching converters. In contrastto numerical techniques, analytic techniques provide analytic expressions representing theoperation and performance of the converters.The most popular continuous-time technique is the small-signal analysis, which uses either circuit averaging [1], state-space averaging [2], or PWM switch modeling [3, 4]. In [1]analytical techniques were developed to represent buck, boost and buck-boost convertersby approximate continuous models. Simple analytical expressions in terms of the circuitcomponents were derived to characterize the low-frequency response of such systems. In[2] the above technique was generalized by introducing the state-space averaging method.The state-space descriptions of each switching mode were replaced by a single state-spacedescription, hence eliminating the switching process from consideration and representingthe average effect of the switched networks during operation cycle. The system was furthersimplified by perturbing the averaged system and then linearizing the resulting perturbedequations around the steady-state values. After a considerable amount of matrix manipulations, the system characteristics such as input impedance, output impedance, line-to-outputtransfer function, and control to-output characteristics, were obtained.1.3 Control Design of Switching DC-DC ConvertersThe converters are required to provide robust behavior and to operate without instabilityunder a variety of operation conditions. Hence the converter system may be subject to thedisturbances of load, input voltage, and system configuration variations.To improve the dynamic performances of converters, closed-loop control is indispensable.Generally, the linear small-signal model obtained using state-averaging and linearizationtechniques around an operating point is adopted for the controller design. However, sincethe model is dependent on the operating conditions and system configuration, the controllerwith fixed parameters (e.g., the PI and optimal controllers) which are adequate under thedesigned condition may not be so for other operating conditions. It is well known that robustcontrol technic is one of the most effective techniques for dealing with parameter variations.Several attempts have been made to apply robust control theory for DC-DC power converters. The linear quadratic Gaussian/loop transfer recovery methodology was used in [5]to design a controller for a series parallel resonant converter. In [6], a controller for a buckboost converter with peak current control was designed using the μ-synthesis procedure. In[7, 8], H approach was applied to design controllers for boost and buck-boost converters.Nonlinear H -control theory has been applied to regulate a PWM Cuk converter underparameter uncertainties and exogenous inputs which generate the reference trajectories [9].In [10], H and μ-synthesis control methods have been applied to Telecom power supplies.But most of the existing robust control techniques are too complex theoretically for practical engineers to understand. It follows that optimal performance is generally not achieved,because traditional control methods design are used in practice.This thesis proposes the use of robust control techniques to derive a controller for dc-dcconverters, which are able to cope with the parameter variations in the converter’s powerstage. In particular, this thesis proposes the use of ”Quantitative Feedback Theory” or QFTapproach [11] which operates on the frequency domain to design a robust output voltagecontroller for switching-mode power converters. It was introduced by Isaac Horowitz in1960s. This technique takes into account the uncertainty that may be present in the processand its environment, and establishes a balance between the complexity of controller and2
Chapter 1Introductioncomplexity of design. It also differs in the way in which uncertainty is characterized asgain-phase variations or templates in the Nichols chart.1.4 Stability and Subsystems InteractionsStability is the most important requirement for switching-mode power supply systems. Theissue of stability is closely related to the EMI filter design for subsystems powered throughswitching power converters. Improper designs of the input filter for such subsystems mayresult in undesirable interactions [12, 13].Considerable interest is focused on evaluating the stability of subsystem interactions indistributed power systems. Usually, the impedance ratio stability criterion suggested in [12]is used to analyze the stability of interactions between two interconnecting subsystems. Forexample, the stability of a spacecraft DC distributed power system is addressed in [14].Stability analysis for a system with a source converter and one or more load converters isgiven in [15, 16]. The design rules are usually based on the separation of the impedancelevels at the interface of the subsystems. After [12, 17], many efforts have been taken indefining less conservative rules, see e.g. [18].All these examples covered relatively simple system configurations, with an ideal voltagesource, one source converter, and one or more load converters with resistive loads. Theexamples represented particular case studies rather than universal analysis tools. The resultswere obtained by tedious analytical developments for a particular system configuration ratherthan applying computer-aided analysis techniques to easily reconfigurable global systemmodel.1.5 Modeling and Dynamics Analysis of Multimodule DC-DC ConvertersAs a viable solution to demanding power requirements, power supplies for distributed powerapplications employ several converter modules in parallel. The resulting multimodule converters offer efficient processing of high current and built-in redundancy [19, 20, 21, 22].However, standard converter modules may not have identical characteristic, which causesunbalance of current sharing. Modules delivering large currents will have their life-timeshortened and the system reliability degraded [21]. Many factors contribute to the fact thatmodules not being identical, such as component tolerances, non-identical electrical conductors connected from the converters to current distribution and so on. Therefore, a unifiedconsistent modeling approach is necessary to understand the dynamic behavior of the powersupply and also to design a controller that regulates the output voltage and achieves balancedcurrent distribution of the converter.With current-sharing control, the output current of a multimodule converter is equallydistributed among parallel modules, thereby improving reliability and reducing current stresson switching devices. Furthermore, the parallel processing of the load current provides faulttolerance to the system against the failure of a single module.1.6 Research ObjectivesObjectives of this thesis can roughly be divided into the following main categories: To give clear physical insight into the concept of switching-mode power converters, andto present a unified modeling methodology of the buck, boost and buck-boost convert3
Chapter 1Introductioners operating in continuous-conduction mode (CCM) and discontinuous-conductionmode (DCM). To implement a robust control approaches (i.e. QFT) in synthesizing robust controllersfor DC-DC switching power converters in order to improve their dynamic performanceby minimizing the effects of load disturbances over the specified region of plant uncertainties. To study the interaction of the subsystems in a distributed power supply system toensure proper overall operation. The aim is to analyze the effect of the input filterand load on the dynamics of the converter. The main purpose is to develop designguidelines which prevent instabilities and performance degradations of the converter. To improve the performance characteristics of multimodule parallel DC-DC convertersystem through modeling, control design and simulation.1.7 Outline of the ThesisThe thesis is organized as follows: In Chapter 2, an overview of a switched-mode supplysystem is given, and the modeling methodology of the system is discussed. In Chapter 3, theQuantitative Feedback Theory (QFT) is applied successfully to design a robust controllerfor DC-DC buck, boost and buck-boost converters operating in continuous conduction mode(CCM) and discontinuous conduction mode (DCM). In Chapter 4, the use of two-portunterminated network representation is demonstrated. Subsystem impedance interactionsand stability analysis for distributed power supply systems are analyzed. The guidelinesfor how to design an optimal input filter for a switching power supply application, whichprevents instabilities and performance degradations of the converter, are presented.In Chapter 5, the small-signal model of DC-DC paralleled converters with individual voltage loop and Master-Slave Control (MSC) circuit is developed using small-signal equivalenttwo-port model. The dynamic characteristics of the current-sharing loop is derived. A robustcurrent-sharing controller which takes into account the stability and ensures distribution ofcurrents among the modules is designed.Conclusions are drawn and some further work considerations are presented in Chapter 6.Appendix A.1 gives some Simulink/SimPower SystemsTM models which have been used togenerate the results presented in the thesis.The following notation is adopted: The capital letters denote the DC values of associatedquantities, the “hatted” small letters the ac or perturbed value excluding the switching ripple,and the small letters denote the total values. In equations the notation for the time variablet is suppressed, when no confusion is possible, e.g. i instead of i(t) etc. The calculus isusually done in Laplace domain, which is not expressed explicitly except in the special caseswhen there is a chance of confusion.1.8 Thesis’s ContributionThe main contributions of this thesis can be summarized as follows: A new application of QFT to the control design of DC-DC switching-mode powerconverters is presented and examined for all basic converters operating in CCM andDCM in VMC and PCMC configurations.4
Chapter 1Introduction The analysis and simulation results show the practical applicability an
parameter variations. In the thesis a robust control design procedure based on the QFT method (Quantitative Feedback Theory) is applied successfully for switching-mode DC-DC converters in order to achieve robust output in spite of different uncertainties. Simulation results are presented to demonstrate and validate the control design, showing good
Mathematical QFT. It intersects with many di erent sub-disciplines including al-gebra, analysis, group theory, measure theory and many others. It di ers from the original QFT in several respects: First, some familiar concepts from the phys-ical theory will not reappear on the mathematics side, as tractable mathematical counterparts are missing.
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instruction method where learners of different levels form small groups and work together towards a specific objective. Learners take the responsibility of their own learning and of those in the group so the success of one member is a success of all members. Piaget (1932, in Webb, 2009: 3) argues that cognitive conflict leads to higher levels of reasoning and learning. When a student notices a .
