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There are two main fields of application of pulse-modulated sys tems, communications and control. Communication is not a subject of our concern in this book. Controlling by a pulse-modulated feed attracted our efforts. The peculiarity of this book is that all back the sampled-data systems are considered in continuous time, so no discrete time schemes are presented. And finally, we pay a little at tention to pulse-amplitude modulation which was treated in a vast number of publications. The primary fields of our interest are pulse width, pulse-frequency, and pulse-phase modulated control systems. The study of such systems meets with substantial difficulties. An engineer, who embarks on theoretical investigations of a pulse-mo dulated control, is often embarrassed by the sophisticated mathe matical tools he needs to know. When a mathematician, who looks for practical applications of his mathematical machinery, meets with these systems, he faces a lot of of complicated technical schemes and terms. Probably this is the reason why publications on pulse modu lation are seldom in scientific journals. As for books on this subject (save on amplitude modulation), the significant part of them is in Russian and hardly available for a non-Russian reader.
The fundamental equation that describes limit cycles in nonlinear sampled-data systems was derived. The equivalence of limit cycles with finite pulsed systems having a periodically varying sampling-rate was observed, and the methods of analysis applied to the latter were extended to obtain these limit cycles with the aid of final value theorem. This fundamental equation is modified and simplified under certain assumptions as it can be applied to systems both with and without integrators. The limitation on the longest period of saturated and unsaturated oscillation is investigated and the critical gain for their existence is derived, starting from the modified fundamental equation. Also, the stability of limit cycles and the equilibrium point is considered, based on Neace's method. Various kinds of non-linearities, namely, pulse-width modulation, relay saturating amplifier with linear zone and quantized level amplifier are discussed. Self-excited oscillations are mainly examined, as well as the possible existence and stability of limit cycles, however, the method can be extended to forced oscillations.