## Doctoral Dissertations

#### Title

Semistate systems and the continuation method

Winter 1991

Dissertation

Engineering

#### Degree Name

Doctor of Philosophy

Paul J Nahin

#### Abstract

This thesis stems from and continues the research of the author and R. Newcomb on semistate theory initiated under the joint Polish-American Program on Active Microelectronic Systems. The main theme is the development of design and analysis techniques through semistate theory, especially for systems containing hysteresis. The use of hysteresis can lead to simplified realizations, for example replacing second order systems by first order ones. Since hysteresis can often be conveniently realized by active electronic components it holds considerable promise for future integrated circuits. However the multivalued nature of hysteresis precludes the use of standard methodologies, such as that of state variable. Semistate theory fills this gap.

We also present a continuation-type method for solving nonlinear semistate systems of equations. Because semistate systems arise naturally in a number of physical situations as for example in the analysis of linear and nonlinear electrical circuits, neural networks, hysteresis, in prescribed path control problems or in constrained robots, there has been recently a considerable interest in solution methods.

The continuation methodology for semistate systems is based on a well recognized and useful topological theory of an embedding model in solving nonlinear nondynamic equations. Since semistate systems generally have nondynamic variables mixed in with dynamic ones, a continuation-type method is proposed as a possible way to solve nonlinear semistate equations. The basic idea is to partition the semistate system into a differential part and an algebraic part and then introduce parameter $\lambda$ and constant linear terms. While the parameter $\lambda$ is slowly varied from 0 to 1 and a series of problems is being solved, the linear terms which dominate at $\lambda$ = 0 disappear at $\lambda$ = 1 thus giving a solution to the original problem.

Examples are given to illustrate not only the feasibility but also to show the efficiency of the method exemplified by rapid convergence and good accuracy with relatively large step sizes. Moreover, our examples prove that the continuation method has been adapted to semistate systems with the significant result that it will converge in some cases where other robust simulators such as SPICE, and PSPICE, will not. Suggestions for extending this research are also presented.

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