7. Chaotic and random point processes: Analysis, design and application to hybrid systems

Processes characterised by point events occurring in time are called point processes. Emissions from a radioactive source, action potential in a nerve fibber, traffic flow passing through a designated location on the highway, are all examples of point processes. In our tutorial we firstly discuss point processes associated with deterministic chaotic dynamics of hybrid systems such as power electronic circuits, systems involving switches, impacting mechanical systems, continuous systems controlled by discrete logic (aircraft auto-pilot modes, thermostats, chemical plants with on-off valves and pumps, automobile automatic transmissions). An understanding of bifurcations (qualitative change in a system's dynamical behaviour) is very important for the operation and control of any system. In this tutorial, we seek to develop a theoretical understanding of the bifurcation phenomena that occur in hybrid systems. We'll show that such systems yield piecewise smooth maps under sampled-data modelling. Next, we'll introduce various smooth and non-smooth bifurcations that can happen in such systems. In presenting the border collision bifurcations, we'll review the various approaches to the problem and present a synthetic theoretical framework that can be readily applied in explaining bifurcation phenomena in hybrid systems. Then we present a systematic approach to the statistical analysis, control and design of hybrid dynamical systems with chaotic behaviour in terms of the statistical properties of the output signal. We introduce the system model and an appropriate signal model and construct correspondences between them. We carry out a statistical analysis of signal models in a form of pulse shaped or filtered renewal processes and a quantized pseudo-Markov processes. We show the advantage to analyse the continuous-time dynamics by means of point processes. Thus the design (or the so-called inverse) problem can be reduced to the analysis and design of chaotic maps with prescribed statistical properties. We review also how to approach the analysis of hybrid systems in the chaotic regime using a state densities approach. We illustrate our methods by practical examples such as DC-DC converters, chaotic bit-stream generators, charge pumps, harmonic and Duffing oscillators. Our tutorial pays a special attention to power electronics, as an application-oriented discipline, which has been developed around some specific power conversion problems in industrial, commercial, residential and aerospace environments. We provide an overview of the key aspects of research done in this area. So, we review the classical methodology for Electro-Magnetic Interference (EMI) reduction based on a frequency modulation of the clock signal (present in digital circuits and power converters) with a sinusoidal or with an optimised periodic signal. Then, we show that further EMI reduction can be achieved by substituting the periodic modulating signals with a chaotic one. Other strategies for interference reduction are spectral shaping based on randomised modulation, and chaotic operation, which flatten the switching spectrum (at the expense of a corresponding broadening). Exploiting the random nature of the chaotic operation of DC-DC converters could provide a way of obtaining a prescribed power spectrum. The lectures will emphasize intuitive reasoning with minimum mathematical details.