#### Document Type

Dissertation

#### Date of Award

Fall 10-31-1973

#### Degree Name

Doctor of Engineering Science in Electrical Engineering

#### Department

Electrical Engineering

#### First Advisor

Andrew Ulrich Meyer

#### Second Advisor

Harlan J. Perlis

#### Third Advisor

Gideon Peyser

#### Fourth Advisor

Mauro Zambuto

#### Abstract

The objective of this dissertation is to study the dynamics of systems consisting of interconnections of an arbitrary number of complete-reset pulse frequency modulators (CRPFM's) and linear dynamical subsystems (in general, time-varying, lumped and/or distributed). CRPFM, which represents a generalization of several types of pulse frequency modulators (PFM's), consists of two basic components; a multi-input dynamic element, called the timing-filter (TF) and a threshold device (TD). Whenever the output of the TF reaches a given threshold value the TD generates an impulse and, at the same time, resets all the states of the TF to zero. This dissertation is devoted to two basic aspects of system motion, namely stability of the equilibrium and periodic operation.

Stability is defined in terms of finiteness of the number of pulses emitted by all modulators. This definition of "finite-pulse stability" (FPS) is related to L_{1} ∩ L_{P} output stability and implies finite energy expended. An improved Lyapunov-like approach is presented which, however, is difficult to employ for higher order systems. A direct criterion for FPS is given which is not only easy to apply, but also provides bounds on the number of pulses emitted by each modulator. A comparison is presented between these criteria and previous stability conditions available for special classes of CRPFM systems (e.g., systems with integral PFM or relaxation PFM). In representative examples, the direct FPS criterion yields comparable (or better) stability regions (of parameters).

The second part is devoted to the study of the basic aspects of "periodic" behavior. For multi-modulator PFM systems, the usual concept of periodicity (or almost periodicty) is not meaningful. Therefore, a weaker concept, that of "ε_{e}-near periodicity" is introduced. This notion involves an observation interval (which is usually finite) and a measure of "desired accuracy" or "observation accuracy". Certain necessary and sufficient conditions for the existence of ε_{e}-near periodic motion are presented. For an IPFM system with a time-invariant linear part, a matrix relationship is given, which relates the "period" and the net number of pulses emitted by each modulator over that period to the system parameters.

Periodic behavior is further investigated on a time-discretized approximation of the CRPFM system which reduces to a system containing ideal delays, summing junctions and threshold elements. However, it is still difficult to obtain analytical results from the resulting (nonlinear) difference equations (except for very short periods of oscillation); nevertheless, these equations can be "linearized" by introduction of extra variables, using Fukunaga's method for nonlinear switching nets. Therefore, classical linear techniques (based on characteristic polynomials and eigenvectors) can be used to obtain information about periodic motion. This approach also applies to McCulloch Pitts type of neural nets and extends existing results on periodic behavior in such networks.

#### Recommended Citation

Gulcur, Halil Ozcan, "Dynamics of interconnected systems with pulse frequency modulators" (1973). *Dissertations*. 1291.

https://digitalcommons.njit.edu/dissertations/1291