Motivations and Perspectives

Abstract

This book deals with the relationship between the qualitative behavior and the mathematical structure of nonlinear, discrete-time dynamic models. The motivation for this treatment is the need for such models in computerized, model-based control of complex systems like industrial manufacturing processes or internal combustion engines. Historically, linear models have provided a solid foundation for control system design, but as control requirements become more stringent and operating ranges become wider, linear models eventually become inadequate. In such cases, nonlinear models are required, and the development of these models raises a number of important new issues. One of these issues is that of model structure selection, which manifests itself in different ways, depending on the approach taken to model development (this point is examined in some detail in Sec. 1.1). This choice is critically important since it implicitly defines the range of qualitative behavior the final model can exhibit, for better or worse. The primary objective of this book is to provide insights that will be helpful in making this model structure choice wisely. One fundamental difficulty in making this choice is the notion of nonlinearity itself: the class of “nonlinear models” is defined precisely by the crucial quality they lack. Further, since much of our intuition comes from the study of linear dynamic models (heavily exploiting this crucial quality), it is not clear how to proceed in attempting to understand nonlinear dynamic phenomena. Because these phenomena are often counterintuitive, one possible approach is to follow the lead taken in mathematics books like Counterexamples in Topology (Steen and Seebach, 1978). These books present detailed discussions of counterintuitive examples, focusing on the existence and role of certain critical working assumptions that are required for the “expected results” to hold, but that are not satisfied in the example under consideration. As a specific illustration, the Central Limit Theorem in probability theory states, roughly, that “sums of N independent random variables tend toward Gaussian limits as TV grows large.” The book Counterexamples in Probability (Stoyanov, 1987) has an entire chapter (67 pages) entitled “Limit Theorems” devoted to achieving a more precise understanding of the Central Limit Theorem and closely related theorems, and to clarifying what these theorems do and do not say.