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UBC Theses and Dissertations

Steady-state ocsillations and stability of on-off feedback systems Mohammed, Auyuab


Methods for studying the behaviour of on-off feedback systems, with the emphasis on steady-state periodic phenomena, are presented in this thesis. The two main problems analyzed are (1) the determination of the periods of self and forced oscillations in single-, double-, and multiloop systems containing an arbitrary number of on-off elements; and (2) the investigation of the asymptotic stability in the small of single-loop systems containing one on-off element which may or may not have a linear region of operation. To study the periodic phenomena in on-off systems, methods of determining the steady-state response of a single on-r-off element are first described. Concepts pertaining to the steady-state behaviour are then introduced: in this respect it has been found that generalizations of the concepts of the Hamel and Tsypkin loci and also of the phase characteristic of Neimark are useful in the study of self and forced oscillations. Both the Tsypkin loci and the phase characteristic concepts are used to determine the possible periods of self and forced oscillations in single- and double-loop systems containing an arbitrary number of on-off elements; these concepts are also applied to multiloop systems. On-off elements containing a linear region of operation, called a proportional band, are then described: both the transient and periodic response are presented. An approximate method for determining the periodic response is given. The concept of the Tsypkin loci is used to determine the possible periods of self and forced oscillations in a single-loop system containing one on-off element with a proportional band. The asymptotic stability in the small, or local stability, of the periodic states of single-loop systems containing one ideal on-off element has been considered by Tsypkin. In this thesis, Tsypkin's results have been generalized to include the cases of on-off elements containing a proportional band. The stability of such systems is determined by the stability of equivalent sampled-data systems with samplers having finite pulse widths. Finally, this stability problem is solved by a direct approach, one that makes use of the physical definition of local stability; the results obtained by this method agree with those derived by the sampled-data approach.

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