Introduction of Chaos in Electric Drive Systems

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Introduction of Chaos in Electric Drive Systems K. T. Chau Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, China Zheng Wang School of Electrical Engineering, Southeast University, Nanjing, China Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Features of Chaos Nonlinearity Determinism Nonlinearity is a necessary, but not sufficient condition for the occurrence of chaos Determinism Chaos must follow one or more deterministic equations that do not contain any random factors Sensitive dependence on initial conditions A small change in the initial state of the system can lead to extremely different behavior in its final state Aperiodicity Chaotic orbits are aperiodic, but not all aperiodic orbits are chaotic Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Chaos in Electronic Circuits The investigation of chaos in electronic circuits can be grouped as: One-dimensional-map circuits (switched-capacitor circuit) Higher-dimensional-map circuits (infinite impulse response digital filter) Continuous-time autonomous circuits (Chua circuit) Continuous-time non-autonomous circuits (phase-locked loop circuit) Fig. 1 Chua circuit. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Chaos in Telecommunications The chaos-based telecommunications can be grouped as: Chaotic masking (feedback-based chaotic masking; observer based chaotic masking) Chaotic modulation (communication based on a modulated transmitter parameter) Chaotic switching (chaos shift keying based chaotic switching; differential chaos shift keying based chaotic switching) Fig. 2 Chaos-based communication system. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Chaos in Power Electronics Analysis of chaos in power converters Buck converter, boost converter, Ćuk converter, et al. Control of chaos in power converters Parameter perturbation, time-delayed feedback control, et al. Application of chaos in power converters: Electromagnetic compatibility improvement, chaotic targeting, et al. Fig. 3 Chaos in simple buck converter. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Chaos in Power Systems The research of chaos in power systems can be grouped as: Power system stability and control Power flow optimization Unit commitment scheduling Load forecasting Fault analysis Fig. 4 Chaos in simple power system. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Chaos in Electric Drive Systems Investigation of chaos in electric drive systems includes: Chaos in DC drive systems Chaos in induction drive systems Chaos in PM brushless AC drive systems Chaos in PM brushless DC drive systems Chaos in synchronous reluctance drive systems Chaos in switched reluctance drive systems Chaos in doubly salient PM drive systems Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Criterion for Chaos Lyapunov exponent Fractal dimensions Entropy The Lyapunov exponent is the exponential rate of divergence and convergence. If the maximum Lyapunov exponent of a dynamical system is positive, this system is chaotic; otherwise, it is nonchaotic. Fractal dimensions The dimension of an attractor is a measure of the number of active variables and the complexity of the equations required to model the system dynamics. If the attractor’s dimension is not an integer, the attractor is a strange attractor. Entropy The sum of the positive Lyapunov exponents is the Kolmogorov-Sinai (K-S) entropy which is a positive constant for a chaotic system. The chaotic degree increases with the value of K-S entropy. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Route to Chaos Period doubling cascade route to chaos The stable fixed points become unstable in a series of period doubling bifurcation, and subharmonic behavior occurs. Intermittency transition to chaos From saddle-node bifurcation From subcritical bifurcation From inverse period doubling bifurcation Chaos from manifold tangle When the manifolds intersect transversely once, they intersect infinite times. It results in stretching and folding actions, and gives an embedded horseshoe map which leads to chaos. Chaos from crisis Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Fundamentals of Electric Drive Systems Fig. 5 Functional block diagram of electric drive system. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Fundamentals of Electric Drive Systems Fig. 6 Classification of electric motors. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

DC Drive Systems Fig. 7 Motor and circuit topologies of DC drive system. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Induction Drive Systems Fig. 8 Motor and circuit topologies of induction drive system. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

PM Brushless Drive Systems Fig. 9 Motor topologies of PM brushless drive system (Left: surface mounted PM; Middle: inset PM; Right: interior PM). Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.

Switched Reluctance Drive Systems Fig. 10 Motor and circuit topologies of switched reluctance drive system. Chaos in Electric Drive Systems: Analysis, Control and Application, First Edition. K.T. Chau and Zheng Wang. © 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd.