1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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.