Oscillations in pulse frequency modulated control systems

This dissertation is concerned with oscillations and their stability which occur in a control system containing two nonlinearities, separated by linear elements. Specifically the nonlinearities are Integral Pulse Frequency Modulators and the linear elements are described by ordinary differential equations which are linear. The IPFM can be modelled by a quantizer with hysteresis and many other PFM laws are related to IPFM, thus the study applies to more than just IPFM alone. Boundaries on the system parameters are identified within which free oscillation may be possible. These boundaries give sufficient conditions for stability and necessary conditions for instability. Also since initial conditions play such an important part in the free motion of this class of systems, certain initial condition zones will be identified. These zones give the initial conditions of the unforced system which will ultimately drive the linear plants to the origin (asymptotic stability). Three types of motion are specifically identified: (1) free oscillation, (2) free periodic oscillation and (3) forced periodic oscillation. Free oscillation, not necessarily periodic, is studied by developing a compound describing function analysis. This type of analysis will be applicable to all systems of the given configuration and some generalizations may be made beyond the IPFM problem. Free periodic motion is very dependent upon the initial condition of the system with many modes of oscillation possible. The solution of this problem involves the solution of a set of transcendental equations and will be carried out using a modified simplex method. The system parameters necessary for forced periodic motion are derived and the possible periods and modes of oscillation identified. The stability of the forced periodic motion is then investigated. The results of this investigation yields a set of matrices, conditions on which, if satisfied, will indicate stability in the small of the periodic motion. Digital and analog computer techniques are used throughout the investigation to verify the theoretical results