Nonlinear Phenomena in Power Electronics Bifurcations, Chaos, Control, and Applications

by Banerjee, Soumitro; Verghese, George C.

• ISBN13: 9780780353831
• ISBN10: 0780353838
• Edition: 1st
• Format: Hardcover
• Copyright: 2001-07-16
• Publisher: Wiley-IEEE Press

Summary

Brings the knowledge of 24 experts in this maturing field out from the narrow confines of academic circles, and makes it accessible to graduate students and power electronics professionals alike. * Provides practicing engineers with the knowledge to predict power requirement behavior. * The insights gained from this all-inclusive compilation will ultimately lead to better design methodologies.

Author Biography

Soumitro Banerjee, Associate Professor, Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, India Soumitro Banerjee has been at the Indian Institute of Technology, in the Department of Electrical Engineering since 1985. He currently teaches courses on 'Dynamics of Physical Systems', 'Signals and Networks', 'Energy Resources and Technology', 'Fractals, Chaos and Dynamical Systems' and 'Nonconventional Electrical Power Generation'. His research interests include bifurcation theory and chaos, and he has written and co-written over 43 papers on these subjects.

Table of Contents

Preface	xv
Acknowledgments	xix
List of Contributing Authors	xxi
Introduction	1	(24)
D. C. Hamill
S. Banerjee
G. C. Verghese
Introduction to Power Electronics	1	(4)
Power Switching Devices	2	(2)
Sources of Nonlinearity in Power Electronics	4	(1)
Power Converters	4	(1)
Electrical Machines and Drives	4	(1)
Power Systems	5	(1)
An Example: The Buck DC/DC Converter	5	(10)
Conventional Model of the Buck Converter	6	(1)
Continuous Conduction Mode	7	(2)
Actual System Behavior	9	(1)
Nonlinear Map-Based Model of the Buck Converter	9	(4)
Discontinuous Conduction Mode	13	(1)
Limitations and Extensions of Average Models	14	(1)
Study of Nonlinear Dynamics and Chaos in Power Electronics	15	(5)
Conclusions	20	(5)
Dynamic Models of Power Converters	25	(412)
Introduction to Power Electronic Converters and Models	25	(13)
G. C. Verghese
A. M. Stankovic
Introduction	25	(1)
Types of Power Electronic Converters	25	(1)
High-Frequency PWM DC/DC Converters	25	(1)
Other High-Frequency PWM Converters	26	(1)
Other Inverters	27	(1)
Resonant Converters	27	(1)
Phase-Controlled Converters	28	(1)
AC/AC Converters	28	(1)
Averaged and Sampled-Data Models for Analysis, Simulation, and Control of Converter Dynamics	28	(2)
Switched State-Space Model for the Boost Converter	30	(2)
Averaged Model for the Boost Converter	32	(1)
Averaged Model for Current-Mode Control of the Boost Converter	33	(1)
Sampled-Data Models for the Boost Converter	34	(2)
Extensions	36	(1)
Generalized Averaging	36	(1)
Generalized State-Space Models	36	(2)
A Closer Look at Sampled-Data Models for Power Converters	38	(15)
F. Vasca
M. di Bernardo
G. Olivar
Introduction	38	(1)
Poincare Maps for Smooth and Nonsmooth Dynamical Systems	38	(2)
Piecewise-Smooth Power Electronic Circuits	40	(3)
Power Electronic Systems as Hybrid Systems	43	(3)
Stroboscopic Maps	46	(2)
Periodic Phase Sequence and Time-Varying Inputs	48	(1)
A Stroboscopic Nonswitching Map	48	(1)
Closed-Loop Maps	48	(1)
Switching Maps	49	(1)
S-Switching Maps for DC/DC Converters	49	(1)
A-Switching Maps for DC/DC Converters	50	(1)
Simplified Maps	51	(1)
Conclusions	51	(2)
Basics of Bifurcation and Chaos Theory
Introduction to Nonlinear Dynamics and Chaos	53	(14)
S. Banerjee
System State, and	53	(2)
State-Space Models
Autonomous Systems and Nonautonomous Systems	55	(1)
Vector Fields of Linear, Linearized, and Nonlinear Systems	55	(2)
Attractors in Nonlinear Systems	57	(2)
Chaos	59	(2)
Poincare Map	61	(1)
Dynamics of Discrete-Time Systems	62	(2)
Fractal Geometry	64	(2)
Lyapunov Exponent	66	(1)
Bifurcation	66	(1)
Bifurcations of Smooth Maps	67	(6)
J. H. B. Deane
The Pitchfork Bifurcation	68	(1)
The Saddle-Node Bifurcation	68	(1)
The Period-Doubling Bifurcation	69	(1)
The Neimark Bifurcation	70	(3)
Bifurcations in Piecewise-Smooth Maps	73	(16)
S. Banerjee
C. Grebogi
The Normal Form	77	(2)
Bifurcations in the One-Dimensional Normal Form	79	(1)
Border Collision Pair Bifurcation	80	(1)
Border-Crossing Bifurcations	81	(1)
Bifurcations in the Two-Dimensional Normal Form	82	(2)
Classification of Border Collision Bifurcations	84	(1)
Border Collision Pair Bifurcation	84	(1)
Border-Crossing Bifurcations	85	(4)
Nonstandard Bifurcations in Discontinuous Maps	89	(5)
I. Dobson
S. Banerjee
The Method of Schwarzian Derivatives	94	(7)
C. K. Tse
Background	94	(1)
Problem Description	94	(1)
Mechanism of Period Doubling	95	(1)
Schwarzian Derivative and Period-Doublings ad infinitum	96	(1)
Application to Power Electronics	97	(1)
Illustrative Example: The Boost Converter	98	(2)
Interpretation and Application of the Result	100	(1)
Remarks and Summary	100	(1)
Coexisting Attractors, Basins of Attraction, and Crises	101	(10)
E. Fossas
G. Olivar
Characteristic (Floquet) Multipliers	101	(1)
Invariant Sets and Invariant Manifolds	102	(1)
Homoclinic and Heteroclinic Orbits	103	(1)
Coexisting Attractors	104	(1)
The Role of Invariant Manifolds and Basins of Attraction	105	(1)
Crises	106	(1)
Interior Crises	107	(1)
Boundary Crises	108	(1)
Two-Dimensional Maps	109	(2)
Experimental and Computational Techniques for Investigation of Nonlinear Phenomena
Techniques of Experimental Investigation	111	(12)
C. K. Tse
Introduction	111	(1)
Overview of Simulation Study and Verification	111	(1)
Experimental Investigation	112	(1)
Displaying Time-Domain Waveforms, Attractors, and Spectra	112	(2)
Displaying Poincare Sections	114	(1)
Principle of Poincare Section Measurement	115	(2)
Example: Free-Running Cuk Converter	117	(1)
Poincare Sections for Nonautonomous Circuits	118	(1)
Displaying Bifurcation Diagrams	118	(1)
Basic Operational Requirements	119	(1)
Digital Implementation and Related Issues	120	(1)
Other Methods, Problems, and Practical Issues	121	(1)
Example: Boost Converter Under Current-Mode Control	121	(2)
Techniques of Numerical Investigation	123	(6)
S. Banerjee
D. C. Hamill
Simulation of Power Electronic Circuits	123	(1)
Problems Arising from Varying Topology	124	(1)
Problems Arising from Incompatible Boundary Conditions	125	(1)
Obtaining Bifurcation Diagrams	125	(1)
Plotting Basins of Attraction in Systems with Multiple Attractors	126	(1)
Computing the Maximal Lyapunov Exponent	127	(2)
Computation of Averages Under Chaos	129	(20)
J. L. Rodriguez Marrero
G. C. Verghese
R. Santos Bueno
S. H. Isabelle
Introduction	129	(1)
Chaotic Operation of DC/DC Converters Under Current-Mode Control	130	(1)
Describing Chaotic Behavior Via Densities	130	(4)
Calculation of the Time-Average of the Inductor Current	134	(1)
Analysis of DC/DC Converters	135	(7)
Average Switching Frequency and Average Duty Ratio	142	(1)
Experimental Results	143	(2)
Conclusions	145	(4)
Calculation of Spectral Peaks in a Chaotic DC/DC Converter	149	(16)
J. H. B. Deane
Characterization of Spectral Properties	149	(2)
Motivation and Outline	151	(1)
The Simplified Mapping	152	(2)
Approximation of the Mean State Variables	154	(1)
The Power Density Spectrum of the Inductor Current	155	(3)
The Invariant Density Algorithm	158	(2)
Practical Results	160	(1)
Experimental Results	160	(3)
Discussion	163	(2)
Computer Methods to Analyze Stability and Bifurcation Phenomena	165	(12)
Y. Kuroe
Introduction	165	(1)
Nonlinear Systems and Stability of Periodic Solutions	165	(2)
Computer Methods to Analyze Stability	167	(2)
Computation of the Jacobian Matrix	169	(2)
Analysis Method Based on Transient Simulator	171	(3)
Computer Method to Analyze Bifurcation Phenomena	174	(1)
Classification of Bifurcations	174	(1)
Method to Determine Bifurcation Values	175	(2)
Computation of Operating-Mode Boundaries	177	(15)
Y. Kuroe
T. Kato
G. C. Verghese
Introduction	177	(1)
What Is Operating-Mode Analysis?	178	(1)
Computation of Operating-Mode Boundary by Curve Tracing	179	(1)
Conditions that Define Operating-Mode Boundaries	179	(2)
Numerical Tracing of Boundary Curves	181	(1)
Computation of Steady-State Sensitivities	182	(1)
Numerical Examples	183	(4)
Computation of Operating-Mode Boundaries by a Binary-Box Method	187	(1)
Basic Approach	187	(1)
Binary-Box Method	188	(1)
Application to More Complicated Converters	188	(4)
Nonlinear Phenomena in DC/DC Converters
Border Collision Bifurcations in the Current-Mode-Controlled Boost Converter	192	(7)
S. Banerjee
P. Ranjan
Modeling and Analysis	192	(3)
Analysis of Bifurcations	195	(4)
Bifurcation and Chaos in the Voltage-Controlled Buck Converter with Latch	199	(9)
S. Banerjee
D. Kastha
S. Das
Overview of Circuit Operation	199	(1)
Experimental Results	200	(2)
Coexisting Attractors and Crises	202	(6)
Routes to Chaos in the Voltage-Controlled Buck Converter without Latch	208	(21)
M. di Bernardo
G. Olivar
F. Vasca
Buck Converter Modeling Under Voltage-Mode Control	208	(1)
Differential equations	209	(1)
Discrete-Time Map and Periodic Orbits	210	(2)
Different Types of Periodic Orbits	212	(1)
Analytical Study of Periodic Orbits: Existence and Stability	212	(5)
One-Dimensional Bifurcation Diagrams	217	(1)
The Main Bifurcation Diagram	217	(2)
Secondary Bifurcations	219	(1)
Chaotic Attractors in the Buck Converter	220	(1)
3T-Periodic Orbits and the Three-Piece Chaotic Attractor	220	(1)
Invariant Manifolds and Basins of Attraction	220	(3)
6T-Periodic Orbits	223	(1)
12T-Periodic Orbits	224	(1)
5T-Periodic Orbits and the Jump to Larger Chaos	225	(4)
Saddle-Node and Neimark Bifurcations in PWM DC/DC Converters	229	(11)
C. C. Fang
E. H. Abed
Introduction	229	(1)
General Sampled-Data Model for Closed-Loop PWM Converters	229	(3)
Periodic Solution Before and After Local Bifurcation	232	(1)
Saddle-Node Bifurcation in Buck Converter Under Discrete-Time Control	232	(2)
Neimark Bifurcation in Buck Converter Under Voltage-Mode Control	234	(2)
Neimark Bifurcation in Buck Converter with Input Filter Under Voltage-Mode Control	236	(2)
Neimark Bifurcation in Buck Converter with Input Filter Under Current-Mode Control	238	(2)
Nonlinear Analysis of Operation in Discontinuous-Conduction Mode	240	(8)
C. K. Tse
Review of Operating Modes	240	(1)
Derivation of Discrete-Time Maps	241	(2)
Period-Doubling Bifurcation	243	(2)
Computer Simulations and Experiments	245	(1)
Remarks and Summary	246	(2)
Nonlinear Phenomena in the Cuk Converter	248	(14)
C. K. Tse
Review of the Cuk Converter and its Operation	248	(1)
Discrete-Time Modeling for Fixed Frequency Operation	249	(2)
Free-Running Current-Mode-Controlled Cuk Converter	251	(1)
Autonomous System Modeling	251	(2)
Dimensionless Equations	253	(1)
Stability of Equilibrium Point and Hopf Bifurcation	254	(2)
Local Trajectories from Describing Equation	256	(2)
Computer Simulation Study	258	(4)
Nonlinear Dynamics in Thyristor and Diode Circuits
I. Dobson
Introduction	262	(1)
Ideal Diode and Thyristor Switching Rules	263	(1)
Static VAR System Example	263	(2)
Poincare Map	265	(2)
Jacobian of Poincare Map	267	(7)
Thyristor Current Function and Transversality	268	(1)
Relations Between On and Off Systems	269	(1)
Derivation of Jacobian Formula	270	(1)
Interval Containing a Switch-On	270	(1)
Interval Containing a Switch-Off	271	(1)
Assembling the Jacobian	272	(1)
Discussion of Jacobian Formula	273	(1)
Switching Damping	274	(4)
Simple Example	274	(1)
Switching Damping in the SVC Example	275	(1)
Variational Equation	276	(2)
Switching Time Bifurcations	278	(8)
Switching Time Bifurcations and Instability	278	(2)
Switching Time Bifurcations for Transients	280	(1)
Misfire Onset as a Transcritical Bifurcation	281	(2)
Noninvertibility and Discontinuity of the Poincare Map	283	(1)
Multiple Attractors and Their Basin Boundaries	284	(2)
Diode Circuits	286	(3)
Transversality and Poincare Map Jacobian Formula	286	(1)
Poincare Map Jacobian for the DC/DC Buck-Boost Converter in Discontinuous Mode	286	(2)
Poincare Map Continuity and Switching Time Bifurcations	288	(1)
Firing Angle Control	289	(3)
Nonlinear Phenomena in Other Power Electronic Systems
Modeling a Nonlinear Inductor Circuit	292	(6)
J. H. B. Deane
Introduction	292	(1)
The Circuit	292	(1)
Saturating and Hysteretic Inductor Modeling	293	(1)
Differential Equation for the Circuit	294	(1)
Results	295	(1)
Bifurcation Diagram Comparison	296	(1)
Poincare Section Comparison	297	(1)
Conclusions	298	(1)
Inverters Under Tolerance Band Control	298	(15)
A. Magauer
Introduction	298	(1)
Functioning Principle	299	(1)
System Model and Equation	300	(2)
Poincare Map	302	(1)
Circuit Realization	303	(1)
Mode of Oscillations, Bifurcations, and Crises	303	(1)
Chaotic Mode	304	(1)
Symmetry-Breaking Bifurcation	305	(2)
Merging Crisis	307	(1)
Interior Crisis	308	(1)
Saddle-Node Bifurcation and Square-Wave Mode	308	(3)
Boundary Crisis	311	(1)
Period-Doubling Bifurcations	312	(1)
Conclusions	312	(1)
Nonlinear Noise Effects in Power Converters	313	(15)
P. T. Krein
P. Midya
Introduction	313	(1)
Discussion of Switching Noise	314	(1)
External Noise Effects in Open-Loop Converters	315	(1)
External Noise Action	315	(1)
Background of Analysis	316	(2)
Summary of Assumptions	318	(1)
Probability Density Function of Switch Timing	318	(1)
Implications of the Non-Gaussian Duty Ratio Distribution with Latch	319	(2)
External Noise Effects in Closed-Loop DC/DC Converters	321	(1)
The Nature of Closed-Loop Noise Effects	321	(1)
The Closed-Loop Process and System Model	321	(1)
Time Domain Noise Analysis	322	(1)
Confirmation	323	(1)
Frequency Domain Analysis	324	(3)
Summary	327	(1)
Nonlinear Phenomena in the Current Control of Induction Motors	328	(10)
I. Nagy
Z. Suto
System Model	329	(1)
Voltage Source Converter (VSC)	329	(1)
AC Side	330	(1)
Hysteresis Current Control (HCC)	331	(1)
Poincare Map	332	(1)
Nonlinear Phenomena	333	(1)
Sensitive Dependence on Initial Condition	333	(1)
Period Doubling Bifurcation	333	(2)
Intermittency	335	(1)
Coexisting Attractors	336	(1)
Numerical Values	336	(1)
Conclusions	337	(1)
Analysis of Stability and Bifurcation in Power Electronic Induction Motor Drive Systems	338	(15)
Y. Kuroe
Introduction	338	(1)
Model of Power Electronic Induction Motor Drive Systems	338	(1)
Model of Induction Motor and its Mechanical Load	338	(2)
Model of Inverter and Rectifier	340	(3)
Poincare Map and Periodicity of Steady States	343	(1)
Case I	343	(1)
Case II	344	(1)
Case III	345	(1)
Stability Analysis	345	(3)
Analysis of Bifurcations	348	(5)
Nonlinear Control and Control of Chaos
Conventional Nonlinear Controls in Power Electronics	353	(4)
P. T. Krein
Introduction	353	(1)
Hysteresis Controllers	354	(1)
Nonlinear Modulation	354	(2)
Multipliers in the Loop	356	(1)
Sliding Mode and Switching Surface Control	357	(14)
P. T. Krein
Introduction	357	(1)
Hysteresis Control	358	(2)
Switching Surface Control Analysis	360	(1)
Trajectories and Equilibria	360	(1)
Switching Surface-Based Control Laws	361	(1)
Necessary Conditions for Switching Surface Controls	362	(1)
Sample Outputs and Hysteresis Design Approaches	363	(2)
Global Stability Considerations	365	(1)
Successor Points	365	(1)
Behavior Near a Switching Surface	366	(1)
Choosing a Switching Surface	367	(2)
Higher Dimensions	369	(1)
Summary	369	(2)
Energy-Based Control in Power Electronics	371	(15)
A. M. Stankovic
G. Escobar
R. Ortega
S. R. Sanders
Introduction	371	(1)
Circuit-Theoretic Approaches	372	(1)
Basic Control	372	(2)
Adaptation	374	(1)
Estimation and Output Feedback	375	(1)
Passivity-Based Control	376	(1)
Basic Controller	376	(2)
Adaptation	378	(1)
Hamiltonian Control	379	(1)
Connections with Sliding-Mode Control	380	(1)
Sliding-Mode Controller Revisited	381	(1)
Passivity-Based Sliding-Mode Controller	382	(1)
Combining SMC with Prediction	383	(1)
Conclusions	384	(2)
Ripple Correlation Control	386	(7)
P. T. Krein
Background	386	(1)
Ripple-Based Control	386	(1)
Ripple Correlation	387	(2)
Some Application Examples	389	(1)
Adaptive Dead Time	389	(1)
Solar Power Processing	390	(1)
Motor Power Minimization in Drives	391	(1)
Summary	392	(1)
Control of Chaos	393	(13)
M. di Bernardo
G. Olivar
C. Batlle
Introduction	393	(1)
A Combination of OGY and Pyragas Methods	394	(1)
Application to the Current-Mode-Controlled Boost Converter	395	(3)
Controlling Border-Collision Bifurcations	398	(1)
Local Feedback Strategy	399	(1)
An Example: A Two-Dimensional Map	400	(1)
Time-Delay Control of Chaos	401	(1)
An Example: TDAS for the Current-Mode Boost Converter	402	(2)
Conclusions	404	(2)
Closed-Loop Regulation of Chaotic Operation	406	(12)
J. L. Rodriguez Marrero
R. Santos Bueno
G. C. Verghese
Introduction	406	(1)
Dynamics and Control	406	(2)
Experimental Results	408	(1)
The OGY Method	409	(2)
Review of the OGY Method	411	(1)
Controlling DC/DC Converters	412	(3)
Integral Control	415	(2)
Conclusions	417	(1)
Control of Bifurcation	418	(10)
C. K. Tse
Y.-M. Lai
Background	418	(1)
Controlling Bifurcation in Discontinuous-Mode Converters	419	(1)
Controlling Bifurcation in Current-Mode-Controlled DC/DC Converters	420	(1)
Use of Compensating Ramp for Controlling Bifurcation	420	(4)
Effects on Dynamical Response	424	(1)
Experimental Measurements	425	(1)
Variable Ramp Compensation	426	(2)
Synchronization of Chaos	428	(9)
C. K. Tse
Background	428	(1)
The Drive-Response Concept	429	(1)
Synchronization in Chaotic Free-Running Cuk Converters	430	(1)
Derivation of the Conditional Lyapunov Exponents	431	(1)
Numerical Calculation of the Conditional Lyapunov Exponents	432	(1)
Computer Simulations	433	(1)
Remarks on Practical Synchronization	433	(4)
Index	437	(4)
About the Editors	441